Special Matrices

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Antisymmetric

see  skew-symmetric .

Bidiagonal

A is upper bidiagonal if a(i,j)=0 unless i=j or i=j-1.
A is lower bidiagonal if a(i,j)=0 unless i=j or i=j+1

A bidiagonal matrix is also tridiagonal, triangular and Hessenberg.

Bisymmetric

A_[n#n] is bisymmetric if it is symmetric about both main diagonals, i.e. if A=A^T=JAJ where J is the exchange matrix.

WARNING: The term persymmetric is sometimes used instead of bisymmetric. Also bisymmetric is sometimes used to mean centrosymmetric and sometimes to mean symmetric and perskewsymmetric.

• A bisymmetric matrix is symmetric, persymmetric and centrosymmetric. Any two of these four properties properties implies the other two.
  • More generally, symmetry, persymmetry and centrosymmetry can each come in four flavours: symmetric, skew-symmetric, hermitian and skew-hermitian. Any pair of symmetries implies the third and the total number of skew and hermitian flavourings will be even. For example, if A is skew-hermitian and perskew-symmetric, then it will also be centrohermitian.
• If A_[2m#2m] is bisymmetric
  • A=[S P^T; P JSJ] for some symmetric S_[m#m] and persymmetric P_[m#m].
  • A is orthogonally similar to [S-JP 0; 0 S+JP]
  • A has a set of 2m orthonormal eigenvectors consisting of m skew-symmetric vectors of the form [u; -Ju]/k and m symmetric vectors of the form [v; Jv]/k where u and v are eigenvectors of S-JP and S+JP respectively and k=sqrt(2).
  • If A has distinct eigenvalues and rank(P)=1 then if the eigenvalues are arranged in descending order, the corresponding eigenvectors will be alternately symmetric and skew-symmetric with the first one being symmetric or skew-symmetric according to whether the non-zero eigenvalue of P is positive or negative.
• If A_[2m+1#2m+1] is bisymmetric
  • A=[S x P^T; x^T y x^TJ; P Jx JSJ] for some symmetric S_[m#m] and persymmetric P_[m#m].
  • A is orthogonally similar to [S-JP 0; 0 y kx^T; 0 kx S+JP] where k=sqrt(2).
  • A has a set of 2m+1 orthonormal eigenvectors consisting of m skew-symmetric vectors of the form [u; 0; -Ju]/k and m+1 symmetric vectors of the form [v; kw; Jv]/k where u and [v; w] are eigenvectors of S-JP and [S+JP kx; kx^T y] respectively and k=sqrt(2).
  • If A has distinct eigenvalues and P=0 then if the eigenvalues are arranged in descending order, the corresponding eigenvectors will be alternately symmetric and skew-symmetric with the first one being symmetric.

Block Diagonal

A is block diagonal if it has the form [A 0 ... 0; 0 B ... 0;...;0 0 ... Z] where A, B, ..., Z are matrices (not necessarily square).

• A matrix is block diagonal iff is the direct sum of two or more smaller matrices.

Centrohermitian

A_[m#n] is centrohermitian if it is rotationally hermitian symmetric about its centre, i.e. if A^T=JA^HJ where J is the exchange matrix.

• Centrohermitian matrices are closed under addition, multiplication and (if non-singular) inversion.

Centrosymmetric

A_[m#n] is centrosymmetric (also called perplectic) if it is rotationally symmetric about its centre, i.e. if A=JAJ where J is the exchange matrix. It is centrohermitian if A^T=JA^HJ and centroskew-symmetric if  A= -JAJ.

• Centrosymmetric matrices are closed under addition, multiplication and (if non-singular) inversion.

Circulant

A circulant matrix, A_[n#n], is a Toeplitz matrix in which a_i,j is a function of {(i-j) modulo n}. In other words each column of A is equal to the previous column rotated downwards by one element.

WARNING: The term circular is sometimes used instead of circulant.

• Circulant matrices are closed under addition, multiplication and (if non-singular) inversion.
• A circulant matrix, A_[n#n] , may be expressed uniquely as a polynomial in C, the cyclic permutation matrix, as A = Sum_i=0:n-1{ a_i,1 Cⁱ} =  Sum_i=0:n-1{ a_1,i C^-i}
• All circulant matrices have the same eigenvectors. If A_[n#n] is a circulant matrix, the normalized eigenvectors of A are the columns of n^-½ F., the discrete Fourier Transform matrix. The corresponding eigenvalues are the discrete fourier transform of the first row of A given by FA^Te₁ = (FA^Ce₁)^C = nF^-1Ae₁ where e₁ is the first comlumn of I.
  • F^-1AF = n^-1 F^HAF=DIAG(FA^Te₁)

Circular

A Circular matrix, A_[n#n], is one for which AA^C = I.

WARNING: The term circular is sometimes used for a circulant matrix.

• A matrix A is circular iff A=exp (j B) where j = sqrt(-1), B is real and exp() is the matrix exponential function.
• If A = B + jC where B and C are real and j = sqrt(-1) then A is circular iff BC=CB and also BB + CC = I.

Companion Matrix

If p(x) is a polynomial of the form a(0) + a(1)*x + a(2)*x² + ... + a(n)*xⁿ then the polynomial's companion matrix is n#n and equals [0 I; -a(0:n-1)/a(n)] where I is n-1#n-1. For n=1, the companion matrix is [-a(0)/a(1)].

The rows and columns are sometimes given in reverse order [-a(n-1:0)/a(n) ; I 0].

• The characteristic and minimal polynomials of a companion matrix both equal p(x).
• The eigenvalues of a companion matrix equal the roots of p(x).

Complex

A matrix is complex if it has complex elements.

Complex to Real Isomporphism

We can associate a complex matrix C_[m#n] with a corresponding real  matrix R_[2m#2n] by replacing each complex element, z, of C by a 2#2 real matrix [z^R -z^I; z^I z^R]=|z|×[cos(t) -sin(t); sin(t) cos(t)] where t=arg(z). We will write C <=> R for this mapping below.

• This mapping preserves the operations +,-,*,/ and, for square matrices, inversion. It does not however preserve • (Hadamard) or ⊗ (Kronecker) products.
• If  C <=> R
  • R = C ⊗ [()^R -()^I; ()^I ()^R] where the operators ()^R and ()^I take the real and imaginary parts respectively.
  • C = (I_[m#m] ⊗ [1 j]) R (I_[n#n] ⊗ [1; 0]) = (I ⊗ [1 0]) R (I ⊗ [1; -j])=½(I ⊗ [1 j]) R (I ⊗ [1; -j]) where j=sqrt(-1).
  • C^R = (I_[m#m] ⊗ [1 0]) R (I_[n#n] ⊗ [1; 0]) = (I_[m#m] ⊗ [0 1]) R (I_[n#n] ⊗ [0; 1])
  • det(R)=|det(C)|²
  • tr(R)=2 tr(C)
  • R is orthogonal iff C is unitary.
  • R is symmetric iff C is hermitian.
    • R is positive definite symmetric iff C is positive definite hermitian.

Vector mapping: Under the isomorphism a complex vector maps to a real matrix: z_[n] <=> Y_[2n#2]. We can also define a simpler mapping, <->, from a vector to a vector as  z_[n] <-> x_[2n] = z ⊗ [()^R; ()^I] = Y [1; 0]

In the results below, we assume z_[n] <-> x_[2n] ,  w_[n] <-> u_[2n] and  C <=> R:

• If w^HCz is known to be real, then w^HCz = u^TRx
  • If  C is hermitian, then,  z^HCz =  x^TRx
•  z^Hz = x^Tx

To relate the martrix and vector mappings, <->  and <=>, we define the following two block-diagonal matrices: E = I_[n#n] ⊗ [0 1; 1 0] and N = I_[n#n] ⊗ [1 0; 0 -1]. We now have the following properties (assuming z_[n] <-> x_[2n]  and  C <=> R):

• E²=N²=I
• E^T=E, N^T=N
• EN=-NE
• ENEN=NENE=-I
• x^TENx=x^TNEx=0
• ENRNE = NEREN = R
  • RNE = NER
  • REN = ENR
• ENREN = NERNE = -R
• C^H <=> R^T
• C^T <=> NR^TN
• C^C <=> NRN
• jC <=> ENR = REN = -NER = -RNE
• z <-> x and z <=> [x ENx]
• z^C <-> Nx and z^C<=> [Nx Ex]
• z^H <-> x^T and z^H <=> Y^T = [x^T; x^TNE]
• z^T <-> x^TN and z^T <=> [x^TN; x^TE]

Convergent

A matrix A is convergent if A^k tends to 0 as k tends to infinity.

• A is convergent iff all its eigenvalues have modulus < 1.
• A is convergent iff there exists a positive definite X such that X-A^HXA is positive definite (Stein's theorem)
• If S_k is defined as I+A+A²+ … +A^k, then A is convergent iff S_k converges as k tends to infinity. If it does converge its limit is (I-A)^-1.

See also: Stability

Cyclic Permutation Matrix

The n#n cyclic permutation matrix (or cyclic shift matrix), C, is equal to [0_n-1^T 1; I_n-1#n-1 0_n-1]. Its elements are given by c_i,j = δ_{i,1+(j mod n)} where δ_i,j is the Kronecker delta.

• C is a toeplitz, circulant, permutation matrix.
• Cx is the same as x but with the last element moved to the top and all other elements shifted down by one position.
• C^-1 = C^T = C^n-1
• Cⁿ =  I

Decomposable

A matrix, A, is fully decomposable (or reducible) if there exists a permutation matrix P such that P^TAP is of the form [B C; 0 D] where B and D are square.
A matrix, A, is partly-decomposable if there exist permutation matrices P and Q such that P^TAQ is of the form [B C; 0 D] where B and D are square.
A matrix that is not even partly-decomposable is fully-indecomposable.

Defective[!]

A matrix, X:n#n, is defective if it does not have n linearly independent eigenvectors, otherwise it is simple.

• X is defective iff A is not diagonalizable.
• X is defective iff the geometric and algebraic multiplicity differ for at least one eigenvalue.
• X is non-defective iff its Jordan form is diagonal.

Derogatory

An n*n square matrix is derogatory if its minimal polynomial is of lower order than n.

Diagonal

A is diagonal if a(i,j)=0 unless i=j.

• Diagonal matrices are closed under addition, multiplication and (where possible) inversion.
• The determinant of a square diagonal matrix is the product of its diagonal elements.
• If D is diagonal, DA multiplies each row of A by a constant while BD multiplies each column of B by a constant.
• If D is diagonal then XDX^T = sum_i(d_i × x_ix_i^T) and XDX^H = sum_i(d_i × x_ix_i^H) [1.15]
• If D is diagonal then tr(XDX^T) = sum_i(d_i × x_i^Tx_i) and tr(XDX^H) = sum_i(d_i × x_i^Hx_i) = sum_i(d_i × |x_i|²) [1.16]
• If D is diagonal then AD = DA iff a_i,j=0 whenever d_i,i != d_j,j. [1.12]
  • If D = DIAG(c₁I₁, c₂I₂, ..., c_MI_M) where the c_k are distinct scalars and the I_k are identity matrices, then AD = DA iff A = DIAG(A₁, A₂, ..., A_M) where each A_k is the same size as the corresponding I_k. [1.13]

The functions DIAG(x) and diag(X) respectively convert a vector into a diagonal matrix and the diagonal of a square matrix into a vector. The function sum(X) sums the rows of X to produce a vector. In the expression below,  •  denotes elementwise (a.k.a. Hadamard) multiplication.

• diag(DIAG(x)) = x
• x^T(diag(Y)) = tr(DIAG(x)Y)
• DIAG(x) DIAG(y) = DIAG(x • y)
• diag(XY) = sum(X • Y^T)

Diagonalizable or Diagonable or Simple or Non-Defective

A matrix, X, is diagonalizable (or, equivalently, simple or diagonable  or non-defective) if it is similar to a diagonal matrix otherwise it is defective.

• If X is diagonalizable, it may be written X=EDE^-1 where D is a diagonal matrix of eigenvalues and the columns of E are the corresponding eigenvectors.
• [X, Y diagonalizable]: The diagonalizable matrices, X and Y, commute, i.e. XY=YX, iff they can be decomposed as X=EDE^-1 and Y=EGE^-1 where D and G diagonal and the columns of E for a common set of eigenvectors.
• The following are equivalent:
  • X is diagonalizable
  •  The Jordan form of X is diagonal.
  • For each eigenvalue of X, the geometric and algebraic multiplicities are equal.
  •  X has n linearly independent eigenvectors.

Diagonally Dominant

A square matrix A_n#n is diagonally dominant if the absolute value of each diagonal element is greater than the sum of absolute values of the non-diagonal elements in its row. That is if for each i we have |a_i,i| > sum_{j != i}(|a_i,j|) or equivalently abs(diag(A)) > ½ABS(A) 1_n#1.

• [Real]: If the diagonal elements of a square matrix A are all >0 and if A and A^T are both diagonally dominant then A is positive definite.
• If A is diagonally dominant and irreducible then
  1. A is non singular
  2. If diag(A) > 0 then all eigenvalues of A have strictly positive real parts.

Discrete Fourier Transform

The discrete fourier transform matrix, F_[n#n], has f_p,q = exp(-2jπ(p-1) (q-1) n^-1).

• Fx is the discrete fourier transform (DFT) of x.
• F is a symmetric, Vandermonde matrix.
• F^-1 = n^-1F^H=n^-1F^C
  • n^-½F is symmetric and unitary.
• If y = Fx then y^Hy = n x^Hx. This is Parseval's theorem.
• F is a Vandermonde matrix.
• det(F) = n^½n.
• tr(F) = 0.
• FC =  DIAG(Fe₂) F where C is the cyclic permutation matrix and e₂ is the second column of I.
• If A_[n#n] is a circulant matrix, the normalized eigenvectors of A are the columns of n^-½ F. The corresponding eigenvalues are the discrete fourier transform of the first row of A given by FA^Te₁ = (FA^Ce₁)^C = nF^-1Ae₁ where e₁ is the first column of I.
• [n=2^k]: F_[n#n] = GP where:
  • P is a symmetric permutation matrix with P = prod_r=1:k(E_k-r ⊗ [ E_r-1 ⊗ [1 0] ; E_r-1 ⊗ [0 1] ] ) where E_s is a 2^s#2^s identity matrix and ⊗ denotes the Kroneker product. If x=0:n-1 then Px consists of the same numbers but arranged in bit-reversed order (e.g. for n=8, Px = [0; 4; 2; 6; 1; 5; 3; 7] ).
  • G = prod_r=1:k(E_r-1 ⊗ [ [1 1] ⊗ E_k-r ; [1 -1] ⊗ W_k-r ]^T) where the diagonal "twiddle factor" matrix is W_s = DIAG(exp(-2^-s j pi  (0:2^s-1))).
  • Calculation of Fx as GPx is the "decimation-in-time" FFT (Fast Fourier Transform) while Fx = F^Tx = PG^Tx is the "decimation-in-frequency" FFT. In each case only O(n log₂n) non-trivial arithmetic operations are required because most of the non-zero elements of the factors of G are equal to ±1.

Doubly-Stochastic

A real non-negative square matrix A is doubly-stochastic if its rows and columns all sum to 1.

See under stochastic for properties.

Essential

An essential matrix, E, is the product E=US of a 3#3 orthogonal matrix, U, and a 3#3 skew-symmetric matrix, S = SKEW(s). In 3-D euclidean space, a translation+rotation transformation is associated with an essential matrix.

• If E=U SKEW(s) is an essential matrix then
  • E=SKEW(Us) U
  • E^TE = (s^Ts) I - ss^T
  • EE^T =  (s^Ts) I - Uss^TU
  • tr(E^TE) = tr(EE^T) = 2s^Ts
• If E is an essential matrix then so are E^T, kE and WEV where k is a non-zero scalar and W and V are orthogonal.
• E is an essential matrix iff rank(E)=2 and EE^TE = ½tr(EE^T)E. This defines a set of nine homogeneous cubic equations.
• E is an essential matrix iff its singular values are k, k and 0 for some k>0.
  • If the singular value decomposition of E is E = Q DIAG([k; k; 0]) R^T, then we can write E = US where U=Q [0 1 0; -1 0 0; 0 0 1] R^T and S = R [0 -k 0; k 0 0; 0 0 0] R^T = SKEW(R [0; 0; k]).
• If E is an essential matrix then A = kE for some k iff Ex × Ax = 0 for all x  where × denotes the vector cross product.

Exchange

The exchange matrix J_[n#n] is equal to [e_n e_n-1 … e₂ e₁] where e_i is the ^ith column of I. It is equal to I but with the columns in reverse order.

• J is Hankel, Orthogonal, Symmetric, Permutation, Doubly Stochastic.
• J² = I
• JA^T, JAJ and A^TJ are versions of the matrix A that have been rotated anti-clockwise by 90, 180 and 270 degrees
• JA, JA^TJ, AJ and A^T are versions of the matrix A that have been reflected in lines at 0, 45, 90 and 135 degrees to the horizontal measured anti-clockwise.
• det(J_n#n) = (-1)^n(n-1)/2 i.e. it equals +1 if n mod 4 equals 0 or 1 and -1 if n mod 4 equals 2 or 3

Givens Reflection

[Real]: A Givens Reflection is an n#n matrix of the form P^T[Q 0 ; 0 I]P where P is any permutation matrix and Q is a matrix of the form [cos(x) sin(x); sin(x) -cos(x)].

• A Givens reflection is symmetric and orthogonal.
• The determinant of a Givens reflection = -1.
• [2*2]: A 2#2 matrix is a Givens reflection iff it is a Householder matrix.

Givens Rotation

[Real]: A Givens Rotation is an n#n matrix of the form P^T[Q 0 ; 0 I]P where P is a permutation matrix and Q is a matrix of the form [cos(x) sin(x); -sin(x) cos(x)].

• A Givens rotation is orthogonal and a Rotation matrix.
• The determinant of a Givens rotation = +1.

Hadamard [!]

An n*n Hadamard matrix has orthogonal columns whose elements are all equal to +1 or -1.

• Hadamard matrices exist only for n=2 or n a multiple of 4.
• If A is an n*n Hadamard matrix then A^TA = n*I. Thus A/sqrt(n) is orthogonal.
• If A is an n*n Hadamard matrix then det(A) = n^n/2.

Hamiltonian

A real 2n*2n matrix, A, is Hamiltonian if KA is symmetric where K = [0 I; -I 0].

See also: symplectic

Hankel

A Hankel matrix has constant anti-diagonals. In other words a(i,j) depends only on (i+j).

• A Hankel matrix is symmetric.
• [A:Hankel] If J is the exchange matrix, then JAJ is Hankel; JA and AJ are Toepliz.
• [A:Hankel] A+B and A-B are Hankel.

Hermitian

A square matrix A is Hermitian if A = A^H, that is A(i,j)=conj(A(j,i))

For real matrices, Hermitian and symmetric are equivalent. Except where stated, the following properties apply to real symmetric matrices as well.

• [Complex]: A is Hermitian iff x^HAx is real for all (complex) x.
• The following are equivalent
  1. A is Hermitian and +ve semidefinite
  2. A=B^HB for some B
  3. A=C² for some Hermitian C.
• Any matrix A has a unique decomposition A = B + jC where B and C are Hermitian: B = (A+A^H)/2 and C=(A-A^H)/2j
• Hermitian matrices are closed under addition, multiplication by a scalar, raising to an integer power, and (if non-singular) inversion.
• Hermitian matrices are normal with real eigenvalues, that is A = UDU^H for some unitary U and real diagonal D.
• A is Hermitian iff x^HAy=x^HA^Hy for all x and y.
• If A and B are hermitian then so are AB+BA and j(AB-BA) where j =sqrt(-1).
• For any complex a with |a|=1, there is a 1-to-1 correspondence between the unitary matrices, U, not having a as an eigenvalue and hermitian matrices, H, given by U=a(jH-I)(jH+I)^-1 and H=j(U+aI)(U-aI)^-1 where j =sqrt(-1). These are Caley's formulae.
  • Taking a=-1 gives U=(I-jH)(I+jH)^-1=(I+jH)^-1(I-jH) and H=j(U-I)(U+I)^-1=j(U+I)^-1(U-I). 

See also: Definiteness, Loewner partial order

Hessenberg

A Hessenberg matrix is like a triangular matrix except that the elements adjacent to the main diagonal can be non-zero.
A is upper Hessenberg if A(i,j)=0 whenever i>j+1. It is like an upper triangular matrix except for the elements immediately below the main diagonal.
A is lower Hessenberg if a(i,j)=0 whenever i<j-1. It is like a lower triangular matrix except for the elements immediately above the main diagonal.

• A symmetric or Hermitian Hessenberg matrix is tridiagonal.
• If A is upper triangular and B is upper Hessenberg then AB is upper Hessenberg.

Hilbert

A Hilbert matrix is a square Hankel matrix with elements a(i,j)=1/(i+j-1).

• The inverse of a hilbert matrix has integer elements.

Homogeneous

If we define an equivalence relation in which X ~ Y iff X = cY for some non-zero scalar c, then the equivalence classes are called homogeneous matrices and homogeneous vectors.

• Multiplication: If X ~ A and Y ~ B, then XY ~ AB
• Addition: If X ~ A and Y ~ B then it is not generally true that X+Y ~ A+B
• The projective space RPⁿ, consists of all non-zero homogeneous vectors from Rⁿ⁺¹.

Householder

A Householder matrix (also called Householder reflection or transformation) is a matrix of the form (I-2vv^H) for some vector v with ||v||=1.

Multiplying a vector by a Householder transformation reflects it in the hyperplane that is orthogonal to v.

Householder matrices are important because they can be chosen to annihilate any contiguous block of elements in any chosen vector.

• A Householder matrix is symmetric and orthogonal.
• Given a vector x, we can choose a Householder matrix P such that Px=[-k 0 0 ... 0]^H where k=sgn(x(1))*||x||. To do so, we choose v = (x + ke₁)/||x + ke₁|| where e₁ is the first column of the identity matrix.  The first row of P equals -k^-1x^T and the remaining rows form an orthonormal basis for the null space of x^T.
• [2*2]: A 2*2 matrix is Householder iff it is a Givens Reflection.

Hypercompanion

The hypercompanion matrix of the polynomial p(x)=(x-a)ⁿ is an n#n upper bidiagonal matrix, H,  that is zero except for the value a along the main diagonal and the value 1 on the diagonal immediately above it. That is, h_i,j = a if j=i, 1 if j=i+1 and 0 otherwise.

If the real polynomial p(x)=(x²-ax-b)ⁿ with a²+4b<0 (i.e. the quadratic term has no real factors) then its Real hypercompanion matrix is a 2n#2n  tridiagonal matrix that is zero except for a at even positions along the main diagonal, b at odd positions along the sub-diagonal and 1 at all positions along the super-diagonal. Thus for odd i, h_i,j = 1 if j=i+1 and 0 otherwise while for even i, h_i,j = 1 if j=i+1, a if j=i and b if j=i-1.

• The characteristic and minimal polynomials of the hypercompanion matrix equal p(x).

Idempotent [!]

P matrix P is idempotent if P² = P . An idempotent matrix that is also hermitian is called a projection matrix.

WARNING: Some people call any idempotent matrix a projection matrix and call it an orthogonal projection matrix if it is also hermitian.

• The following conditions are equivalent
  1. P is idempotent
  2. P is similar to a diagonal matrix each of whose diagonal elements equals 0 or 1.
  3. 2P-I is involutary.
• If P is idempotent, then:
  • rank(P)=tr(P).
  • The eigenvalues of P are all either 0 or 1. The geometric multiplicity of the eigenvalue 1 is rank(P).
  • P^H, I-P and I-P^H are all idempotent.
  • P(I-P) = (I-P)P = 0.
  • Px=x iff x lies in the range of P.
  • The null space of P equals the range of I-P. In other words Px=0 iff x lies in the range of I-P.
  • P is its own generalized inverse, P^#.
• [A: n#n, F,G: n#r] If A=FG^H where F and G are of full rank, then A is idempotent iff G^HF = I.

Identity[!]

The identity matrix , I, has a(i,i)=1 for all i and a(i,j)=0 for all i !=j

Impotent

A non-negative matrix T is impotent if min(diag(Tⁿ)) = 0 for all integers n>0 [see potency].

Incidence

An incidence matrix is one whose elements all equal 1 or 0.

Integral

An Integral matrix is one whose elements are all integers.

• The following types of matrix are integral: Incidence, Hadamard.

Involutary (also written Involutory)

An Involutary matrix is one whose square equals the identity.

• A is involutary iff ½(A+I) is idempotent.
• A_[2#2] is involutary iff A = +-I or else A = [a  b; (1-a²)/b  -a] for some real or complex a and b.

Irreducible

see under Reducible

Jacobi

see under Tridiagonal

Monotone

A matrix, A, is monotone iff  A^-1 is non-negative, i.e. all its entries are >=0.

In computer science a matrix is monotone if its entries are monotonically non-decreasing as you move away from the main diagonal along either a row or column.

Nilpotent [!]

A matrix A is nilpotent to index k if A^k = 0 but A^k-1 != 0.

• The determinant of a nilpotent matrix is 0.
• The eigenvalues of a nilpotent matrix are all 0.
• If A is nilpotent to index k, its minimal polynomial is t^k.

Non-negative

see under positive

Normal

A square matrix A is normal if A^HA = AA^H

• A_n#n is normal iff any of the following equivalent conditions is true
  • A is unitarily similar to a diagonal matrix.
  • A has an orthonormal set of n eigenvectors
  • eig(A)^Heig(A) = ||A||_F² where ||A||_F is the Frobenius norm.
• The following types of matrix are normal: diagonal, hermitian, skew-hermitian and unitary.
• A normal matrix is hermitian iff its eigenvalues are all real.
• A normal matrix is skew-hermitian iff its eigenvalues all have zero real parts.
• A normal matrix is unitary iff its eigenvalues all have an absolute value of 1.
• For any X_m#n, X^HX and XX^H are normal.
• The singular values of a normal matrix are the absolute values of the eigenvalues.
• [A: normal] The eigenvalues of A^H are the conjugates of the eigenvalues of A and have the same eigenvectors.
• Normal matrices are closed under raising to an integer power and (if non-singular) inversion.
• If A and B are normal and AB=BA then AB is normal.

Orthogonal [!]

A real square matrix Q is orthogonal if Q'Q = I. It is a proper orthogonal matrix if det(Q)=1 and an improper orthogonal matrix if det(Q)=-1.

For real matrices, orthogonal and unitary mean the same thing. Most properties are listed under unitary.

Geometrically: Orthogonal matrices in 2 and 3 dimensions correspond to rotations and reflections.

• The determinant of an orthogonal matrix equals +-1 according to whether it is proper or improper.
• Q is a proper orthogonal matrix iff Q = exp(K) or K=ln(Q) for some real skew-symmetric K.
• A 2#2 orthogonal matrix is either a Givens rotation or a Givens reflection according to whether it is proper or improper.
• A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether  it is proper or improper.
• For a=+1 or a=-1, there is a 1-to-1 correspondence between real skew-symmetric matrices, K, and orthogonal matrices, Q, not having a as an eigenvalue given by Q=a(K-I)(K+I)^-1 and K=(aI+Q)(aI-Q)^-1. These are Caley's formulae.
  • For a=-1 this gives Q=(I-K)(I+K)^-1 and K=(I-Q)(I+Q)^-1. Note that (I+K) is always non-singular.

Permutation

A square matrix P is a permutation matrix if its columns are a permutation of the columns of I.

• A permutation matrix is orthogonal and doubly stochastic.
• The set of permutation matrices is closed under multiplication and inversion.1
• If P is a permutation matrix:
  • P^-1 =  P^T
  • P² = I iff P is symmetric
• P is a permutation matrix iff each row and each column contains a single 1 with all other elements equal to 0.

Persymmetric

A matrix A_[n#n] is persymmetric if it is symmetric about its anti-diagonal, i.e. if A=JA^TJ where J is the exchange matrix. It is perhermitian if A=JA^HJ and perskewsymmetric if  A= -JA^TJ.

WARNING: The term persymmetric is sometimes used for a bisymmetric matrix.

• If A is persymmetric then so is A^k for any positive or, providing A is non-singular, negative k.
• A Toeplitz matrix is persymmetric.

Polynomial Matrix

A polynomial matrix of order p is one whose elements are polynomials of a single variable x. Thus A=A(0)+A(1)x+...+A(p)x^p where the A(i) are constant matrices and A(p) is not all zero.

See also regular.

Positive

A real matrix is positive if all its elements are strictly > 0.
A real matrix is non-negative if all its elements are >= 0.

• [Perron's theorem] If A_n#n  is positive with spectral radius r, then the real positive value r is an eigenvalue with the following properties:
  • the eigenvalue r is simple, i.e. its algebraic and geometric multiplicities equal 1.
• the eigenvector, x, satisfying Ax = rx can be chosen to have strictly positive real elements.
• the eigenvector, y, satisfying A^Ty = ry  can be chosen to have strictly positive real elements.
• all other eigenvalues have magnitude strictly less than r and their corresponding eigenvectors cannot be chosen to have all elements strictly positive and real.
• The rank-1 impotent matrix, T = xy^T/x^Ty, is the projection onto the eigenspace spanned by x. The limit, lim_m->inf(r^-1A)^m = T = xy^T/x^Ty.
• [Perron-Frobenius theorem] If A_n#n  is irreducible and non-negative with spectral radius r, then the real positive value r is an eigenvalue with the following properties:
  • the eigenvalue r is simple, i.e. its algebraic and geometric multiplicities equal 1.
• the eigenvector, x, satisfying Ax = rx can be chosen to have strictly positive real elements.
• the eigenvector, y, satisfying A^Ty = ry  can be chosen to have strictly positive real elements.
• the eigenvectors associated with any other eigenvalue cannot be chosen to have all elements strictly positive and real.
• If there are h eigenvalues of magnitude r, then these eigenvalues are simple and are given by r exp(2jπk/h) for k=0, 1, …, h-1. h is the period.

Positive Definite

see under definiteness

Primitive

If k is the eigenvalue of a matrix A_n#n having the largest absolute value, then A is primitive if the absolute values of all other eigenvalues are < |k|.

• If A_n#n is non-negative then A is primitive iff A^m is positive for some m>0.
• If A_n#n is non-negative and primitive then lim_m->inf(r^-1A)^m = xy^T where r is the spectral radius of A and x and y are positive eigenvectors satisfying Ax = rx,  A^Ty = ry and x^Ty = 1.

Projection

A projection matrix (or orthogonal projection matrix) is a square matrix that is hermitian and idempotent: i.e. P^H=P²=P.

WARNING: Some people call any idempotent matrix a projection matrix and call it an orthogonal projection matrix if it is also hermitian.

• If P is a projection matrix then P is positive semi-definite.
• I-P is a projection matrix iff P is a projection matrix.
• X(X^HX)^#X^H is a projection whose range is the subspace spanned by the columns of X.
  • If X has full column rank, we can equivalently write X(X^HX)^-1X^H
  • xx^H/x^Hx is a projection onto the 1-dimensional subspace spanned by x.
• If P and Q are projection matrices, then the following are equivalent:
  1. P-Q is a projection matrix
  2. P-Q is positive semidefinite
  3. ||Px|| >= ||Qx|| for all x.
  4. PQ=Q
  5. QP=Q
• [A: idempotent] A is a projection matrix iff ||Ax|| <= ||x|| for all x.

Quaternion

Quaternions are a generalization of complex numbers. A quaternion  consists of a real component and three independent imaginary components and is written as r+xi+yj+zk where i²=j²=k²=ijk=-1. It is approximately true that whereas the polar decomposition of a complex number has a magnitude and 2-dimensional rotation, that of a quaternion has a magnitude and a 3-dimensionl rotation (see below). Quaternions form a division ring rather than a field because although every non-zero quaternion has a multiplicative inverse, multiplication is not in general commutative (e.g. ij=-ji=k). Quaternions are widely used to represent three-dimensional rotations in computer graphics and computer vision as an alternative to orthogonal matrices with the following advantages: (a) more compact, (b) possible to interpolate, (c) does not suffer from "gimbal lock", (d) easy to correct for drift due to rounding errors.

We can represent a quaternion either as a real 4-vector q_R=[r x y z]^T or a complex 2-vector q_C=[r+jy  x+jz]^T. This gives  r+xi+yj+zk = [1 i j k]q_R  = [1 i]q_C. We can also represent it as a real 4#4 matrix Q_R=[r -x -y -z; x r -z y; y z r -x; z -y x r] or a complex 2#2 matrix Q_C=[r+jy -x+jz; x+jz r-jy]. Both the real and the complex quaternion matrices obey the same arithmetic rules as quaternions, i.e. the quaternion matrix representing the result of applying +, -, * and / operations to quaternions is the same as the result of applying the same operations to the corresponding quaternion matrices. Note that q_R=Q_R[1 0 0 0]^T and q_C=Q_C[1 0]^T; we can also define the inverse functions Q_R=QUAT_R(q_R) and Q_C=QUAT_C(q_C). Note that the real and complex representations given above are not the only possible choices.

In the following, P_R=QUAT_R(p_R), Q_R=QUAT_R(q_R), K=DIAG([-1 1 1 1]) and q_R=[r x y z]^T=[r; w]. P_C,p_C,Q_C and q_C are the corresponding complex quantities; the subscripts R and C are omitted below for results that apply to both real and complex representations. 

• The magnitude of the quaternion is m=|q|=sqrt(r²+x²+y²+z²). . A unit quaternion has m = 1. 
  • det(Q_R)=m⁴. det(Q_C)=q^Hq=m²
  • Any quaternion may be written as m times a unit quaternion.
• Q^-1=(q^Hq)^-1Q^H is the reciprocal of the quaternion.
• Q^H is the conjugate of the quaternion; this corresponds to reversing the signs of x, y and z.
• PQ=QUAT(Pq) and P+Q=QUAT(p+q). This illustrates that we may often use the quaternion vectors rather than the matrices when performing arithmetic with a resultant saving in computation.
• P_Rq_R=KQ_R^TKp_R. Note however that KQ_R^TK is not a quaternion matrix unless Q is a multiple of I (i.e. the corresponding quaternion is purely real).
• (Q_RK)²=(KQ_R)²
• (P_RQ_RK)²=(P_RK)²(Q_RK)²
• Q_R=rI+[0 -w^T; w SKEW(w)]
• [|q|=1] (Q_RK)²=(KQ_R)²=[1 0; 0 S] where S is a 3#3 rotation matrix corresponding to an angle of 2cos^-1(r) about an axis whose unit vector is w/sqrt(1-r²).
  • Every 3#3 rotation matrix  corresponds to a unit quaternion matrix that is unique except for its sign, i.e. +Q and -Q correspond to the same rotation matrix. Thus the decomposition of a quaternion into a magnitude and 3-dimensional rotation is only invertible to within a sign ambiguity.
  • [|p|=|q|=1] If (P_RK)²==[1 0; 0 R] and (Q_RK)²==[1 0; 0 S], then (P_RK)²(Q_RK)²=(P_RQ_RK)²=[1 0; 0 RS]. This shows that multiplying unit quaternions is equivalent to multiplying rotation matrices but may be more efficient computationally if it is possible to use quaternion vectors rather than matrices for intermediate results.

Rank-one

A non-zero matrix A is a rank-one matrix iff it can be decomposed as A=xy^T.

• If A=xy^T is a rank-one matrix then
  •  If A=pq^T then p=kx and q=y/k for some scalar k. That is, the decomposition is unique to within a scalar multiple.
• If A=xy^T is a square rank-one matrix then
  • A has a single non-zero eigenvalue equal to x^Ty=y^Tx. The associated right and left eigenvectors are respectively x and y.
• Frobenius Norm: ||A||_F²=tr(A^HA)=x^Hx×y^Hy
• Pseudoinverse: A⁺=A^H/ ||A||_F² =A^H/tr(A^HA)=A^H/(x^Hx×y^Hy) where  ||A||_F is the Frobenius Norm.

Reducible

A matrix A_n#n is reducible (or fully decomposable) if if there exists a permutation matrix P such that P^TAP is of the form [B C; 0 D] where B and D are square. As a special case 0_1#1 is regarded as reducible. A matrix that is not reducible is irreducible.

WARNING: The term reducible is sometimes used to mean one that has more than one block in its Jordan Normal Form.

• An irreducible matrix has at least one non-zero off-diagonal element in each row and column.
• A_n#n is irreducible iff (I + ABS(A))^n-1 is positive.

Regular

A polynomial matrix, A, of order p is regular if det(A) is non-zero.

• An n#n square polynomial matrix, A(x), of order p is regular iff det(A) is a polynomial in x of degree n*p.

Rotation Matrix

[Real]: A Rotation matrix, R, is an n*n matrix of the form R=U[Q 0 ; 0 I]U^T where U is any orthogonal matrix and Q is a matrix of the form [cos(x) -sin(x); sin(x) cos(x)]. Multiplying a vector by R rotates it by an angle x in the plane containing u and v, the first two columns of U. The direction of rotation is such that if x=90 degrees, u will be rotated to v. 

• A Rotation matrix is orthogonal with a determinant of +1.
• All but two of the eigenvalues of R equal unity and the remaining two are exp(jx) and exp(-jx) where j is the square root of -1. The corresponding unit modulus eigenvectors are [u v][1 -j]^T/sqrt(2) and [u v][1 +j]^T/sqrt(2).
• R=I+(cos(x)-1)(uu^T+vv^T)+sin(x)(vu^T-uv^T) where u and v are the first two columns of U. 
  • If x=90 degrees then R=I-uu^T-vv^T+vu^T-uv^T .
  • If x=180 degrees then R=I-2uu^T-2vv^T 
  • If x=270 degrees then R=I-uu^T-vv^T-vu^T+uv^T
• [3#3] R = ww^T+cos(x)(I-ww^T)+sin(x)SKEW(w) = I+sin(x)SKEW(w)+(1-cos(x))SKEW(w)² where the unit vector w = u × v is the axis of rotation. [See skew-symmetric for the definition and properties of SKEW()].
  • tr(R) = 2 cos(x) + 1
  • Every 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether its determinant is +1 or -1.
  • The product of two 3#3 rotation matrices is a rotation matrix.
  • A 3#3 rotation matrix may be expressed as the product of three rotations about the x, y and z axes respectively. The corresponding rotation angles are the Euler angles. The order in which the rotations are performed is significant and is not standardised. Using Euler angles is often a bad idea because their relation to the rotation axis direction is not continuous.
  • R=(I-K)(I+K)^-1 where K=-tan(x/2)*SKEW(w) except when x=180 degrees. This is the Caley transform.
  • If x=90 degrees then R=ww^T+SKEW(w) =(I+SKEW(w))(I-SKEW(w))^-1
  • If x=180 degrees then R=2ww^T-I
  • If x=270 degrees then R=ww^T-SKEW(w)=(I-SKEW(w))(I+SKEW(w))^-1
  • ADJ(R-I)=2(1-cos(x))ww^T where ADJ() denotes the adjoint. All columns of this rank-1 matrix are multiples of w.
  • Every 3#3 rotation matrix  corresponds to a quaternion matrix that is unique except for its sign.

Shift Matrix

A  shift matrix, or lower shift matrix, Z,  is a matrix with ones below the main diagonal and zeros elsewhere.
Z^T has ones above the main diagonal and zeros elsewhere and is an upper shift matrix.

• ZA, Z^TA, AZ, AZ^T, ZAZ^T are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.
• Z_n#n is nilpotent.

Signature

A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.

• A signature matrix is involutary.

Simple

An n*n square matrix is simple (or, equivalently, diagonalizable or diagonalizable  or non-defective) if all its eigenvalues are regular, otherwise it is defective.

• A matrix is simple iff it is similar to a diagonal matrix.

Singular

A matrix is singular if it has no inverse.

• A matrix A is singular iff det(A)=0.

Skew-Hermitian

A square matrix K is Skew-Hermitian (or antihermition) if K = -K^H, that is a(i,j)=-conj(a(j,i))

For real matrices, Skew-Hermitian and skew-symmetric are equivalent. The following properties apply also to real skew-symmetric matrices.

• S is Hermitian iff jS is skew-Hermitian where j = sqrt(-1)
• K is skew-Hermitian iff x^HKy = -x^HK^Hy for all x and y.
• Skew-Hermitian matrices are closed under addition, multiplication by a scalar, raising to an odd power and (if non-singular) inversion.
• Skew-Hermitian matrices are normal.
• If K is skew-hermitian, then K² is hermitian.
• The eigenvalues of a skew-Hermitian matrix are either 0 or pure imaginary.
• Any matrix A has a unique decomposition A = S + K where S is Hermitian and K is skew-hermitian.
• K is skew-hermitian iff K=ln(U) or U=exp(K) for  some unitary U .
• For any complex a with |a|=1, there is a 1-to-1 correspondence between the unitary matrices, U, not having a as an eigenvalue and skew-hermitian matrices, K, given by U=a(K-I)(I+K)^-1 and K=(aI+U)(aI-U)^-1. These are Caley's formulae.
  • Taking a=-1 gives U=(I-K)(I+K)^-1 and K=(I-U)(I+U)^-1. 

   

Skew-Symmetric[!]

A square matrix K is skew-symmetric (or antisymmetric) if K = -K^T, that is a(i,j)=-a(j,i)

For real matrices, skew-symmetric and Skew-Hermitian are equivalent. Most properties are listed under skew-Hermitian .

• Skew-symmetry is preserved by congruence.
• The diagonal elements of a skew-symmetric matrix are all 0. [1.10]
• The rank of a real or complex skew-symmetric matrix is even. [1.11]
• [Real] The non-zero eigenvalues of a real skew-symmetric matrix are all purely imaginary and occur in complex conjugate pairs.
  • If K is skew-symmetric, then I - K is non-singular
• [Real] If A is skew-symmetric, then x^TAx = 0 for all real x.
• [Real] If a=+1 or a=-1, there is a 1-to-1 correspondence between real skew-symmetric matrices, K, and those orthogonal matrices, Q, not having a as an eigenvalue given by Q=a(K-I)(K+I)^-1 and K=(aI+Q)(aI-Q)^-1 . These are Caley's formulae.
• K is real skew-symmetric iff K=ln(Q) or Q = exp(K) for some real proper orthogonal matrix Q. 
• [Real 3#3] All 3#3 skew-symmetric matrices have the form SKEW(a) =  [0 -a₃ a₂; a₃ 0 -a₁; -a₂ a₁ 0] for some vector a.
  • SKEW(ka) = k SKEW(a) for any scalar k
  • The vector cross product is given by a × b = SKEW(a) b = -SKEW(b) a
  • SKEW(a) b = 0 iff a = kb for some scalar k
  • SKEW(a)²ⁿ=(-a^Ta)^n-1aa^T+(-a^Ta)ⁿI=(-a^Ta)^n-1(aa^T-(a^Ta)I) for integer n>=1
    • SKEW(a)²=aa^T-(a^Ta)I
  • SKEW(a)²ⁿ⁺¹=(-a^Ta)ⁿSKEW(a) for integer n>=0
    • SKEW(a)³=-(a^Ta)SKEW(a)
  • The eigenvalues of SKEW(a) are 0 and +-sqrt(-a^Ta)
    • The eigenvector associated with 0 is ka
    • [Real a]: Eigenvalues are 0 and +-j|a| where j is sqrt(-1). Unless q=r=0 a suitable pair of eigenvectors are [-q²-r² jr-pq pr-jq]^T and [-q²-r² -jr-pq pr+jq]^T.
  • The singular values of SKEW(a) are |a|, |a| and 0.
    • If z=|a| and w=[z₂ z₃]^T, then a singular value decomposition is SKEW(a)=USV^T where U=[z^T; w  I+(z₁-1)^-1ww^T]J, S=DIAG(|a|, |a|, 0) and V=U [0 1 0; -1 0 0; 0 0 1] where J is the exchange matrix (i.e. I with the column order reversed). All other decompositions may be obtained by postmultiplying both U and V by DIAG(Q_[2#2], 1) for some orthogonal Q and/or negating the final column of one or both of U and V.
  • SKEW(a)^T SKEW(a) = SKEW(a) SKEW(a)^T = |a|² I - aa^T
    • tr( SKEW(a)^T SKEW(a))=2a^Ta
  • det([a b c]) = a^T SKEW(b) c = b^T SKEW(c) a = c^T SKEW(a) b, this is the scalar triple product.
    • a^T SKEW(b) a = a^T SKEW(a) b = b^T SKEW(a) a = 0 for all a and b
  • SKEW(a)SKEW(b) = ba^T-(b^Ta)I
  • SKEW(a)SKEW(b) c = (a^Tc)b - (a^Tb)c, this is the vector triple product.
  • For any a and B_[3#3],
    • B^TSKEW(Ba)B = det(B) * SKEW(a)
    • SKEW(Ba)B = ADJ(B)^T SKEW(a) where ADJ(B) denotes the adjoint matrix.
      • [det(B)!=0]: SKEW(Ba) = det(B) * B^-TSKEW(a)B^-1
  • SKEW(SKEW(a)b) = ba^T - ab^T
  • [U orthogonal] The product E = U SKEW(a) = SKEW(Ua) U is an essential matrix
    •  E^TE = (a^Ta) I - aa^T
    •  tr(E^TE) = 2a^Ta.

Sparse

A matrix is sparse if it has relatively few non-zero elements.

Stability

A Stability or Stable matrix is one whose eigenvalues all have strictly negative real parts.
A semi-stable matrix is one whose eigenvalues all have non-positive real parts.

See also: Convergent

Stochastic

A real non-negative square matrix A is stochastic if all its rows sum to 1.. If all its columns also sum to 1 it is Doubly Stochastic.

• All eigenvalues of A are <= 1.
• 1 is an eigenvalue with eigenvector [1 1 ... 1]^T

Sub-stochastic

A real non-negative square matrix A is sub-stochastic if all its rows sum to <=1.

Subunitary

A is subunitary if ||AA^Hx|| = ||A^Hx|| for all x. A is also called a partial isometry.

The following are equivalent:

1. A is subunitary
2. A^HA is a projection matrix
3. AA^HA = A
4. A⁺ = A^H

• A is subunitary iff A^H is subunitary iff A⁺ is subunitary.
• If A is subunitary and non-singular than A is unitary.

Symmetric[!]

A square matrix A is symmetric if A = A^T, that is a(i,j) = a(j,i).

Most properties of real symmetric matrices are listed under Hermitian .

• [Real]: If A is real, symmetric, then A=0 iff x^TAx = 0 for all real x.
• [Real]: A real symmetric matrix is orthogonally similar to a diagonal matrix.
  • [Real, 2#2] A=[a b; b d]=RDR^T where D is diagonal and R=[cos(t) -sin(t); sin(t) cos(t)] and t=½tan^-1(2b/(a-d)).
• A is symmetric iff it is congruent to a diagonal matrix.
• Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
• Any symmetric matrix A can be expressed as A=UDU^T where U is unitary and D is real, non-negative and diagonal with its diagonal elements arranged in non-increasing order (i.e. d_i,i <= d_j,j for i < j). This is the Takagi decomposition and is a special case of the singular value decomposition.

See also Hankel.

Symmetrizable

A real matrix, A, is symmetrizable if A^TM = MA for some positive definite M.

Symplectic

A matrix, A_[2n#2n], is symplectic if A^HKA=K where K is the antisymmetric orthogonal matrix [0 I; -I 0].

• A is symplectic iff A^-1=K^TA^HK
  • If a symplectic matrix A=[P Q; R S] where P,Q,R,S are all n#n, then A^-1=[S^H -R^H ; -Q^H P^H]
• The set of symplectic matrices of size 2n#2n is closed under multiplication and inversion and so forms a multiplicative group.
• A is symplectic iff it preserves the symplectic form x^HKy, that is (Ax)^HK(Ay) = x^HKy for all x and y. This is analogous to the way that a unitary matrix, U, preserves the inner product: (Ux)^H(Uy)=x^Hy.

See also: hamiltonian

Toeplitz

A toeplitz matrix, A, has constant diagonals. In other words a_i,j depends only on i-j.
We define A=TOE(b_[m+n-1])_[m#n] to be the m#n matrix with a_i,j = b_i-j+n. Thus, b is the column vector formed by starting at the top right element of A, going backwards along the top row of A and then down the left column of A.
In the topics below, J is the exchange matrix.

• A toeplitz matrix is persymmetric and so, if it exists, is its inverse. A symmetric toeplitz matrix is bisymmetric.
• If A and B are toeplitz, then so are A+B and A-B. Note that AB and A^-1 are not necessarily toeplitz.
• If A is toeplitz, then A^T, A^H and JAJ are Toeplitz while JA,  A^TJ,  AJ and JA^T are Hankel.
  • If A_[n#n] is toeplitz, then JA^TJ=(JAJ)^T=A while JA=A^TJ and AJ=JA^T are Hankel.
• TOE(a+b) = TOE(a) + TOE(b)
• TOE(b_[m+n-1])_[m#n]=TOE(Jb)_[n#m]^T
  • TOE(b_[2n-1])_[n#n]=TOE(Jb)_[n#n]^T
• If the lower triangular matrices A_[n#n]=TOE([0_[n-1]; p_[n]]) and B_[n#n]=TOE([0_[n-1]; q_[n]]) then:
  • Aq = Bp = conv(p,q)_1:n
  • AB = BA = TOE([0_[n-1]; Aq]) = TOE([0_[n-1]; Bp]) = TOE([0_[n-1]; conv(p,q)_1:n])
  • A^-1 and B^-1 are toeplitz lower triangular if they exist.
• If the upper triangular matrices A_[n#n]=TOE([ p_[n]; 0_[n-1]]) and B_[n#n]=TOE([ q_[n]; 0_[n-1]]) then:
  • Aq = Bp = conv(p,q)_n:2n-1
  • AB = BA = TOE([Aq; 0_[n-1]]) = TOE([Bp; 0_[n-1]]) = TOE([conv(p,q)_n:2n-1; 0_[n-1]])
  • A^-1 and B^-1 are toeplitz lower triangular if they exist.
• The product TOE(a)_[m#r]TOE(b)_[r#n] is toeplitz iff a_r+1:r+m-1b_1:n-1^T = a_1:m-1b_r+1:r+n-1^T  [1.21]. This m-1#n-1 rank-one matrix identity is equivalent to requiring one of the following conditions:
  1. Both a_r+1:r+m-1=ka_1:m-1 and b_r+1:r+n-1=kb_1:n-1 for the same scalar k. Note that a_1:m-1 and a_r+1:r+m-1 will overlap if m>r+1 and similarly for b if n>r+1.
  2. • For TOE(a) to be square and symmetric, a_1:m-1 must be either symmetric or antisymmetric with k=+1 or -1 respectively (a similar condition applies to TOE(b)).
  3. Either  a_r+1:r+m-1= 0 or b_1:n-1 = 0  and also either a_1:m-1= 0 or b_r+1:r+n-1= 0 . If m=r=n then this condition is equivalent to requiring that A and B are either both upper triangular or both lower triangular or else one of them is diagonal.

Some special cases of this are:

• TOE(a)_[m#r]TOE(b)_[r#n] is toeplitz if a_r+1:r+m-1 = a_1:m-1 and b_r+1:r+n-1= b_1:n-1. Note that this does not make the matrices symmetrical even for square matrices because a_1:m-1 goes backwards along the top row of the matrix.
• TOE([0_[m-1]; a_[r]])_[m#r]TOE([0_[n-1]; b_[r]])_[r#n] = TOE([0_[n+m-r-1]; conv(a,b)_1:r])
• TOE([a_[r]; 0_[m-1]])_[m#r]TOE([b_[r]; 0_[n-1]])_[r#n]  = TOE([ conv(a,b)_r:2r-1; 0_[n+m-r-1]])

• If A=TOE(b)_[m#n]  then JAJ=TOE(Jb)_[m#n]
• TOE([0_[n-p]; a_[m]; 0_[q-m]])_[q-p+1#n] b_[n] = TOE([0_[m-p]; b_[n]; 0_[q-n]])_[q-p+1#m] a_[m] = conv(a,b)_p:q provided that p<=m,n<=q and conv(a,b)_i is taken to be 0 for i outside the range 1 to m+n-1.
  • TOE(a_[m])_[m-n+1#n] b_[n] = conv(a,b)_n:m
  • TOE([0_[n-p]; a_[n]])_[n-p+1#n] b_[n] = TOE([0_[n-p]; b_[n]])_[n-p+1#n] a_[n] = conv(a,b)_p:n
    • TOE([0_[n-1]; a_[n]])_[n#n] b_[n] = TOE([0_[n-1]; b_[n]])_[n#n] a_[n] = conv(a,b)_1:n
  • TOE([a_[n]; 0_[q-n]])_[q-n+1#n] b_[n] = TOE([b_[n]; 0_[q-n]])_[q-n+1#n] a_[n] = conv(a,b)_n:q
    • TOE([a_[n]; 0_[n-1]])_[n#n] b_[n] = TOE([b_[n]; 0_[n-1]])_[n#n] a_[n] = conv(a,b)_n:2n-1
  • TOE([0_[n-1]; a_[m]; 0_[n-1]])_[m+n-1#n] b_[n] = TOE([0_[m-1]; b_[n] ;0_[m-1]])_[m+n-1#m] a_[m] = conv(a,b)
• A symmetric toeplitz matrix is of the form S_[n#n] = TOE([Ja_[n]; 0_[n-1]]+[0_[n-1]; a_[n]])
  • JSJ = S
  • Sb = (TOE([b_[n]; 0_[n-1]])_[n#n]J+TOE([0_[n-1]; b_[n]])_[n#n])a . The matrix on the right is the sum of a lower triangular toeplitz and an upper triangular hankel matrix.

Triangular

A is upper triangular if a(i,j)=0 whenever i>j.
A is lower triangular if a(i,j)=0 whenever i<j.
A is triangular iff it is either upper or lower triangular.
A triangular matrix A is strictly triangular if its diagonal elements all equal 0.
A triangular matrix A is unit triangular if its diagonal elements all equal 1.

• [Real]: An orthogonal triangular matrix must be diagonal
• [n*n]: The determinant of a triangular matrix is the product of its diagonal elements.
• If A is unit triangular then inv(A) exists and is unit triangular.
• A strictly triangular matrix is nilpotent .
• The set of upper triangular matrices are closed under multiplication and addition and (where possible) inversion.
• The set of lower triangular matrices are closed under multiplication and addition and (where possible) inversion.

Tridiagonal or Jacobi

A is tridiagonal or Jacobi if A(i,j)=0 whenever |i-j|>1. In other words its non-zero elements lie either on or immediately adjacent to the main diagonal.

• A is tridiagonal iff it is both upper and lower Hessenberg.

Unitary

A complex square matrix A is unitary if A^HA = I. A is also sometimes called an isometry.

A real unitary matrix is called orthogonal .The following properties apply to orthogonal matrices as well as to unitary matrices.

• Unitary matrices are closed under multiplication, raising to an integer power and inversion
• U is unitary iff U^H is unitary.
• Unitary matrices are normal.
• U is unitary iff ||Ux|| = ||x|| for all x.
• The eigenvalues of a unitary matrix all have an absolute value of 1.
• The determinant of a unitary matrix has an absolute value of 1.
• A matrix is unitary iff its columns form an orthonormal basis.
• U is unitary iff U=exp(K) or K=ln(U) for  some skew-hermitian K.
• For any complex a with |a|=1, there is a 1-to-1 correspondence between the unitary matrices, U, not having a as an eigenvalue and skew-hermitian matrices, K, given by U=a(K-I)(I+K)^-1 and K=(aI+U)(aI-U)^-1. These are Caley's formulae.
  • Taking a=-1 gives U=(I-K)(I+K)^-1 and K=(I-U)(I+U)^-1. 

Vandermonde

An Vandermonde matrix, V_[n#n], has the form [1 x x^•2 … x^•n-1] for some column vector x. (where x^•2 denotes elementwise squaring). A general element is given by v(i,j) = (x_i)^j-1. All elements of the first column of the matrix equal 1. Vandermonde matrices arise in connection with fitting polynomials to data.

WARNING: Some authors define a Vandermonde matrix to be either the transpose or the horizontally flipped version of the above definition.

Vectorized Transpose Matrix

The vectorized transpose matrix, TVEC(m,n), is the mn#mn  permutation matrix whose i,j^th element is 1 if j=1+m(i-1)-(mn-1)floor((i-1)/n) or 0 otherwise.

For clarity, we write T_m,n = TVEC(m,n) in this section.

• [A_[m#n]] (A^T): = T_m,nA: [see vectorization, R.6]
• T_m,n is a permutation matrix and is therefore orthogonal.
• T_1,n = T_n,1 = I
• T_n,m = T_m,n^T = T_m,n^-1
• [A_[m#n], B_[p#q]] B ⊗ A = T_p,m (A  ⊗  B) T_n,q
  • [A_[m#n], B_[p#q]] (A ⊗ B) T_n,q  = T_m,p (B ⊗ A)
  • [a_[n], B_[p#q]] (a  ⊗  B)  = T_n,p (B ⊗ a)

Zero

The zero matrix, 0, has a(i,j)=0 for all i,j

• [Complex]: A=0 iff x^HAx = 0 for all x .
• [Real]: If A is symmetric, then A=0 iff x^TAx = 0 for all x.
• [Real]: A=0 iff x^TAy = 0 for all x and y.
• A=0 iff A^HA = 0