The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

Another
triangular form
Hermite form
0 0.29 0.5
0 -0.87 + 0.29s 0.5 + 0.5s
-1 -0.29s^2 -0.5s^2
If A( s)
has not full row rank then T ( s)
is in staircase form. Similarly, an upper-right
triangular (row staircase) form is achieved by row (unimodular) operations. It results
from the call tri(A,'row').
If B( s)
is a square polynomial matrix with nonsingular constant term then another
upper-triangular form may be obtained by the overloaded macro lu:
B = [ 1 1 s; s+1 0 s; 1-s s 2+s]
B =
1 1 s
1 + s 0 s
1 - s s 2 + s
[V,T] = lu(B)
V =
T =
1 0.33s 0.11s
1 + s 1 + 1.3s + 0.33s^2 0.44s + 0.11s^2
1 - s 1 - 1.7s - 0.33s^2 1 - 0.56s - 0.11s^2
1 1 + 0.33s 0.78s + 0.11s^3
0 -1 -0.67s - s^2 + 0.33s^3
0 0 2 + 2s + 2s^2 - s^3
The triangular forms described above are by no means unique. A canonical triangular
form is called the Hermite form. An n m polynomial matrix A( s)
of rank r is in
column Hermite form if it has the following properties:
it is lower triangular
the diagonal entries are all monic
each diagonal entry has higher degree than any entry on its left
in particular, if the diagonal element is constant then all off-diagonal elements in
the same row are zero
if n r then the last n r columns are zero
The nomenclature in the literature is not consistent. Some authors (in particular
Kailath, 1980) refer to this as the row Hermite form. The polynomial matrix A is in
row Hermite form if it is the transpose of a matrix in column Hermite form. The
command
H = hermite(A)
returns the column Hermite form
H =
1 0 0
1 + s s^2 0
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