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

is lower triangular, and its eigenvalues are for all values of , , and .

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

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

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 to Jordan canonical form by means of a similarity transformation, zero any off-diagonal elements, and transform back [332].

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