Chapter 5

Question 1

Definition: eigenvalues and eigenspaces

Answer 1

Let A be a square matrix.

i) A scalar gamma is called an eigenvalue of A if there is a nontrivial solution to the equation Ax = gamma x
ii) A nonzero vector x such that Ax = gamma x is called an eigenvector of A (corresponding to the eigenvalue gamma)
iii) The set of all solutions to Ax = gamma x for a particular gamma is called the eigenspace of A corresponding to gamma

Question 2

Important stuff about eigenvalues, vectors and spaces

Answer 2

The number gamma is an eigenvalue of A
if and only if Ax = gamma x has nontrivial solutions
if and only if Ax - gamma x = 0 vector has nontrivial solutions
if and only if (A - gamma * Identity)x = 0 has nontrivial solutions
if and only if A - gamma * Identity is not invertible!!!

The eigenspace of A corresponding to the eigenvalue gamma is the solution set for the homogeneous system (A- gamma * Identity)x = 0 vector, i.e.
Nul (A- gammaIdentity). The basis for this eigenspace is the basis for Nul(A-gammaIdentity).

Question 3

Theorem 5.1

Answer 3

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

Question 4

Definition: characteristic equation

Answer 4

Let A be an n x n matrix. The scalar equation det(A-gammaIdentity) = 0 is called the characteristic equation of A. The nth degree polynomial det(A-gammaIdentity) is the characteristic polynomial of A
A scalar gamma is an eigenvalue of A iff gamma satisfies the characteristic equation.
The multiplicity of the eigenvalue gamma is its multiplicity as a zero of the characteristic polynomial.
An n x n matrix has at most n eigenvalues, counting multiplicity.

Question 5

Theorem 5.2

Answer 5

If v1, … , vf are eigenvectors that correspond to distinct eigenvalue gamma1, … , gammaT of a matrix A, then { v1, … , vf } is linearly independent.

Question 6

Definition: diagonalizable

Answer 6

A square matrix A is called 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.

Question 7

The Diagonalization Theorem 5.5

Answer 7

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.
In fact, A = PDP^-1 for some diagonal matrix D if and only if the columns of P are n linearly independent eigenvectors of A and the entries on the main diagonal of D are their corresponding eigenvalues.

Question 8

Theorem 5.6

Answer 8

An n x n matrix with n distinct eigenvalues is diagonalizable