### Eigenspace and Multiplicity

##### 2. Algebraic and geometric multiplicities

In this section, we analyse the diagonalization process in detail.

To diagonalize an n by n matrix A, we first find the eigenvalues. As a root of the characteristic polynomial p(λ) = det(A - λI), any eigenvalue λi has a multiplicity miA defined by (see here for more details)

p(λi) = (λi - λ)miA s(λ) for integer miA > 0 and another polynomial s satisfying s(λi) ≠ 0.

Basically, miA is the number of times λi appears as a root of the characteristic polynomial. This multiplicity miA is called the algebraic multiplicity of the eigenvalue λi. Moreover, this fact about polynomial and the fact that the top degree term of det(A - λI) is (-1)nλn tell us the following.

Let λ1, λ2, ..., λk be all the distinct eigenvalues of A, with algebraic multiplicities m1A, m2A, ..., mkA. Then

det(A - λI) = (λ1 - λ)m1A (λ2 - λ)m2A ... (λk - λ)mkA.

In particular, m1A + m2A + ... + mkA = n.

In the second step of the diagonalization process, we compute a basis Bi = {vi1, vi2, ..., vimiG} for the eigenspace Eλi = nul(A - λiI) of each eigenvalue λi. The number of vectors in Bi is miG = dimEλi, called the geometrical multiplicity of the eigenvalue λi.

Example Based on the computations in this example, this example, and this example, we have the following algebraic and geometric multiplicities.

matrix characteristic
polynomial
eigenvalue algebraic
multiplicity
eigenvector geometric
multiplicity
 [ 13 -4 ] -4 7
(5 - λ)(15 - λ) λ1 = 5 m1A = 1 (1,2) m1G = 1
λ2 = 15 m2A = 1 (-2, 1) m2G = 1
 [ 1 3 -3 ] -3 7 -3 -6 6 -2
(4 - λ)2(- 2 - λ) λ1 = 4 m1A = 2 (1, 1, 0), (1, 0, -1) m1G = 2
λ2 = -2 m2A = 1 (1, 1, 2) m2G = 1
 [ 2 3 -3 ] 0 2 -3 0 0 1
(2 - λ)2(1 - λ) λ1 = 2 m1A = 2 (1, 0, 0) m1G = 1
λ2 = 1 m2A = 1 (-6, 3, 1) m2G = 1

We observe that the algebraic multiplicity is always no smaller than the geometric multiplicity: miAmiG. Moreover, the first two matrices are diagonalizable, for which we have miA = miG. However, the third matrix is not diagonalizable, for which we see the equality does not always hold. The observation will be generalized in the next part.