The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
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Bases and null<br />
spaces<br />
ans =<br />
1<br />
confirms that P has full rank.<br />
<strong>The</strong> normal rank of a polynomial matrix P( s)<br />
equals<br />
max sC P( s)<br />
rank<br />
Similar definitions apply to the notions of normal column rank and normal row rank.<br />
<strong>The</strong> rank is calculated by<br />
rank(P)<br />
ans =<br />
3<br />
As <strong>for</strong> constant matrices, rank evaluation may be quite sensitive and an ad hoc<br />
change of tolerance (which may be included as an optional input parameter) may be<br />
helpful <strong>for</strong> difficult examples.<br />
A polynomial matrix is nonsingular if it has full normal rank.<br />
issingular(P)<br />
ans =<br />
0<br />
<strong>The</strong>re are two important subspaces (more precisely, submodules) associated with a<br />
polynomial matrix A( s):<br />
its null space and its range (or span). <strong>The</strong> (right) null space<br />
is defined as the set of all polynomial vectors x( s)<br />
such that A( s) x( s)<br />
0 . For matrix<br />
A = P(1:2,:)<br />
A =<br />
1 s s^2<br />
1 + s s 1 - s<br />
the (right) nullspace is computed by<br />
N = null(A)<br />
N =<br />
0.35s - 0.35s^2 - 0.35s^3<br />
-0.35 + 0.35s + 0.35s^2 + 0.35s^3<br />
-0.35s^2<br />
Here the null space dimension is 1 and its basis has degree 3.
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