Chapter 5

5.1 Eigenvectors and Eigenvalues

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = λx for some scalar λ. A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ.

The eigenvalues of a triangular matrix are the entries on its main diagonal.

If v₁, ..., v_r are eigenvectors that correspond to distinct eigenvalues λ₁, ..., λ_r of an n x n matrix A, then the set {v₁, ..., v_r} is linearly independent.

5.2 The Characteristic Equation

Let A be an n x n matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling), and let r be the number of such row interchanges. Then the determinant of A, written as det A, is (-1)^r times the product of the diagonal entries u₁₁, ..., u_nn. If A is invertible, then the determinant is nonzero because the diagonal entries are all pivots.

Properties of Determinants
Let A and B be n x n matrices.
a. A is invertible if and only if det A is not 0.
b. det AB = (det A)(det B).
c. det A^T = det A.
d. If A is triangular, then det A is the product of the entries on the main diagonal of A.
e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor.

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the characteristic equation det(A-λI) = 0.

If A is an n x n matrix, then det(A-λI) is a polynomial of degree n called the characteristic polynomial of A. In general, the multiplicity of an eigenvalue λ is its multiplicity as a root of the characteristic equation.

If A and B are n x n matrices, then A is similar to B if there is an invertible matrix P such that P^-1AP = B, or, equivalently, A = PBP^-1. Changing A into P^-1AP is called a similarity transformation.

If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues(with the same multiplicities).

5.3 Diagonalization

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, if A = PDP^-1 for some invertible matrix P and some diagonal matrix D.

The Diagonalization Theorem
An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A = PDP^-1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A the correspond to, respectively, to the eigenvectors in P. In otherwords, A is diagonalizable if and only if there are enough eigenvectors to form a basis for Rⁿ. We call such a basis an eigenvector basis of Rⁿ.

An n x n matrix with n distinct eigenvalues is diagonalizable.

Let A be an n x n matrix whose distinct eigenvalues are λ₁, ..., λ_p.
a. For 1 <= k <= p, the dimensions of the eigenspace for λ_k is less than or equal to the multiplicity of he eigenvalue λ_k.
b. The matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n, and this happens if and only if (i) the characteristic polynomial factors completely into linear factors and (ii) the dimension of the eigenspace for each λ_k equals the multiplicity of λ_k.
c. If A is diagonalizable and B_k is a basis for the eigenspace corresponding to λ_k for each k, then the total collection of vectors in the set B₁, ..., B_p forms an eigenvector basis for Rⁿ.