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

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 [

329].

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