Matrix Relations

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Square matrices A and B are congruent if there exists a non-singular X such that B= XTAX . Congruence is an equivalence relation.

For Hermitian congruence, see Conjuctivity.

Congruence implies equivalence.


Square matrices A and B are conjunctive or hermitely congruent or star-congruent if there exists a non-singular X such that B= XHAX. Conjunctivity is an equivalence relation.


Two matrices are conjugate iff they are similar.

Direct Sum

The Direct sum of matrices A, B, ... is written AB⊕... = DIAG(A, B, ...).


Two m#n matrices, A and B, are equivalent iff there exists a non-singular m#m matix, M, and a non-singular n#n matrix, N, with B=MAN. Equivalence is an equivalence relation.

Hadamard (or Schur) Product

The Hadamard product of two m#n matrices A and B, written in this website AB, is formed by the elementwise multiplication of their elements. The matrices must be the same size.…

Kronecker Product

The Kronecker product of A[m#n] and B[p#q], written AB or KRON(A,B), is equal to the mp#nq matrix [a(1,1)Ba(1,n)B ; • ; a(m,1)Ba(m,n)B ]. It is also known as the direct product or tensor product of A and B. The Kronecker Product operation is often denoted by a • sign enclosed in a circle which we approximate with ⊗. Note that in general AB != BA. In the expressions below a : suffix denotes vectorization.

In the identities below, In = I[n#n] and Tm,n = TVEC(m,n) [see vectorized transpose]

Kroneker Sum

The Kronecker sum of two square matrices, A[m#m] and B[n#n], is equal to  (AIn) +  (Im ⊗  B). It is sometimes written AB but in these pages, this notation is reserved for the direct sum.

Loewner Partial Order

We can define a partial order on the set of Hermitian matrices by writing A>=B iff A-B is positive semidefinite and A>B iff A-B is positive definite.

Orthogonal Similarity

Real square matrices A and B are orthogonally similar if there exists an orthogonal Q such that B= QTAQ .

Orthogonal similarity implies both similarity and congruence.

See also: Unitary similarity


Square matrices A and B are similar (also called conjugate) if there exists a non-singular X such that B=X-1AX . Similarity is an equivalence relation, i.e. it is reflexive, symmetric and transitive.

Similar matrices represent the same linear transformation in a different basis. Similarity implies equivalence .

Unitary Similarity or Unitary Equivalence

Square matrices A and B are unitarily similar if there exists a unitary Q such that B= QHAQ . Unitary similarity is an equivalence relation and implies both similarity and conjunctivity.

This page is part of The Matrix Reference Manual. Copyright © 1998-2022 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "".
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