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Triangular Feedback Matrices

An interesting class of feedback matrices, also explored by Jot [216], 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

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 [329].

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Most General Lossless Feedback Matrices