Special Matrices: Examples

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In this document 2#2 matrices are illustrated by showing the image of the
unit disc under the transformation **y=Ax**. The disc quadrants are coloured
white, red, green and blue.

The value of **x ^{T}Ax** is plotted in polar form for

- |
**A**| equals the*area*of the disc image. - The
*singular values*of**A**are the semi-major and semi-minor axes of the disc image. - The
*eigenvalues*are the lengths of the*eigenvector*lines. - The matrix is posi
*tive definite, negative definite or indefinite*according to whether the**x**curve is all brown, all cyan or a bit of both.^{T}Ax

A defective *n#n* matrix does not have *n* independent
eigenvectors.

- [1 1; 0 1]

- [1 1; 1 -1]

- [1 0.5; 0 0]
- [0.12 0.66; 0.16 0.88]

- [1 0; 0 1]

- [0 1; 0 0]

Orthogonal 2#2 matrices consist of a rotation or a reflection:

- [0.8 -0.6; 0.6 0.8]

- [0.5 0.5; 0.7 0.3]

- [0 0.7; -0.7 0]

For symmetric matrices the eigenvalues and singular values have equal magnitudes and the eigenvectors lie on the axes of the disc image:

- [1 0.3; 0.3 0.7]
- [1 0.6; 0.6 -0.7]

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 "imperial.ac.uk".

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