Symbolic Jordan canonical form of a matrix.
Example:
A = sym ([6 5 -2 -3; -3 -1 3 3; 2 1 -2 -3; -1 1 5 5]);
jordan (A)
⇒ ans = (sym 4×4 matrix)
⎡2 1 0 0⎤
⎢ ⎥
⎢0 2 0 0⎥
⎢ ⎥
⎢0 0 2 1⎥
⎢ ⎥
⎣0 0 0 2⎦
We can also compute the generalized eigenvectors:
[V, J] = jordan (A)
⇒ V = (sym 4×4 matrix)
⎡4 1 5 0⎤
⎢ ⎥
⎢-3 0 -3 1⎥
⎢ ⎥
⎢2 0 1 0⎥
⎢ ⎥
⎣-1 0 1 0⎦
⇒ J = (sym 4×4 matrix)
⎡2 1 0 0⎤
⎢ ⎥
⎢0 2 0 0⎥
⎢ ⎥
⎢0 0 2 1⎥
⎢ ⎥
⎣0 0 0 2⎦
A*V - V*J
⇒ ans = (sym 4×4 matrix)
⎡0 0 0 0⎤
⎢ ⎥
⎢0 0 0 0⎥
⎢ ⎥
⎢0 0 0 0⎥
⎢ ⎥
⎣0 0 0 0⎦
The generalized eigenvectors are the columns of V. Those corresponding to a Jordan block form a cycle. We can check those columns corresponding to the leftmost Jordan block:
lambda = J(2, 2)
⇒ lambda = (sym) 2
B = A - lambda*eye (4);
v2 = V(:, 2)
⇒ v2 = (sym 4×1 matrix)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
v1 = B * v2
⇒ v1 = (sym 4×1 matrix)
⎡4 ⎤
⎢ ⎥
⎢-3⎥
⎢ ⎥
⎢2 ⎥
⎢ ⎥
⎣-1⎦
v0 = B * v1
⇒ v0 = (sym 4×1 matrix)
⎡0⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
See also: @sym/charpoly, @sym/eig.
Package: symbolic