math111_logo Polynomials of Matrix

2. Generalized eigenspace

Let λ1, λ2, ..., λk be all the distinct eigenvalues of an n by n matrix A, with algebraic multiplicities m1, m2, ..., mk. Then by this result, the characteristic polynomial is

p(t) = det(A - λI) = (λ1 - λ)m1 (λ2 - λ)m2 ... (λk - λ)mk.

By Cayley-Hamilton theorem, we have

p(A) = (λ1I - A)m1 (λ2I - A)m2 ... (λkI - A)mk = O.

In particular, we have

In an earlier example, we considered the following diagonalizable matrices

A = [ 13 -4 ], with eigenvalues 5, 15;
-4 7
B = [ 1 3 -3 ], with eigenvalues 4, -2.
-3 7 -3
-6 6 -2

For the polynomial p(t) = (5 - t)(15 - t) = 75 - 20t + t2, applying the matrix p(A) = 75I - 20A + A2 to the basis of eigenvectors of A is the same as multiplying p(λ) = p(5 or 15) = 0 to the eigenvectors. In other words, applying p(A) to the basis vectors always give us


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