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.



Defective

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


Hadamard


Idempotent


Identity


Nilpotent


Orthogonal

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


Stochastic


Skew-Symmetric


Symmetric

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



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