Matrix Reference Manual
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 xTAx is plotted in polar form for x lying on the unit circle (i.e. the locus of xTAx * x is plotted). Positive values are plotted in brown and negative in cyan. When x is an eigenvector, this curve will always coincide with the edge of the disc image. The eigenvectors (if any) are plotted in brown or cyan according to the sign of the corresponding eigenvalue.

• |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 positive definite, negative definite or indefinite according to whether the xTAx curve is all brown, all cyan or a bit of both.

### Defective

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

• [1 1; 0 1]

• [1 1; 1 -1]

### Idempotent

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

• [1 0; 0 1]

• [0 1; 0 0]

### Orthogonal

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

• [0.8 -0.6; 0.6 0.8]

### Stochastic

• [0.5 0.5; 0.7 0.3]

### Skew-Symmetric

• [0 0.7; -0.7 0]

### Symmetric

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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