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Triangular Feedback Matrices

An interesting class of feedback matrices, also explored by Jot [217], is that of triangular matrices. A basic fact from linear algebra is that triangular matrices (either lower or upper triangular) have all of their eigenvalues along the diagonal.4.13 For example, the matrix

$\displaystyle \mathbf{A}_3 = \left[\begin{array}{ccc}
\lambda_1 & 0 & 0\\ [2pt]
a & \lambda_2 & 0\\ [2pt]
b & c & \lambda_3
\end{array}\right]
$

is lower triangular, and its eigenvalues are $ (\lambda_1,
\lambda_2,\lambda_3)$ for all values of $ a$ , $ b$ , and $ c$ .

It is important to note that not all triangular matrices are lossless. For example, consider

$\displaystyle \mathbf{A}_2 = \left[\begin{array}{cc} 1 & 0 \\ [2pt] 1 & 1 \end{array}\right]
$

It has two eigenvalues equal to 1, which looks lossless, but a quick calculation shows that there is only one eigenvector, $ [0,1]^T$ . This happens because this matrix is a Jordan block of order 2 corresponding to the repeated eigenvalue $ \lambda=1$ . A direct computation shows that

$\displaystyle \mathbf{A}_2^n = \left[\begin{array}{cc} 1 & 0 \\ [2pt] n & 1 \end{array}\right]
$

which is clearly not lossless.

One way to avoid ``coupled repeated poles'' of this nature is to use non-repeating eigenvalues. Another is to convert $ \mathbf{A}$ to Jordan canonical form by means of a similarity transformation, zero any off-diagonal elements, and transform back [332].


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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