05 Eigenvalues and eigenvectors

Question 1

eigenvector and eigenvalue

Answer 1

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = λx for som 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.

Question 2

The eigenspace of A corresponding to λ consists of the zero vector and _____.

Answer 2

The eigenspace of A corresponding to λ consists of the zero vector and all the eigenvectors corresponding to λ.

Question 3

λ is an eigenvalue of an n x n matrix A if and only if the equation

_____

has a nontrivial solution.

Answer 3

λ is an eigenvalue of an n x n matrix A if and only if the equation

(A - λI)x = 0

has a nontrivial solution.

Question 4

The eigenvalues of a triangular matrix are the entries on ____.

Answer 4

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

Question 5

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the characteristic equation

_______

Answer 5

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

Question 6

Similarity

Answer 6

If A and B are n x n matrices, then A is imilar to B if there is an invertible matrix P such that P^-1AP = B.

Question 7

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

Answer 7

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

Question 8

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

Answer 8

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 som diagonal matrix D.

Question 9

An n x n matrix A is diagonalizable if and only if A has _____ eigenvectors.

Answer 9

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.

In fact, A = PDP^-1 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 that correspond, respectively, to the eigenvectors in P.

Question 10

A is in other words diagonalizable if and only if there are enough eigenvectors to form ______.

Answer 10

A is in other words diagonalizable if and only if there are enough eigenvectors to form a basis of R^n. We call such a basis an eigenvector basis of R^n.

Question 11

Steps to diagonalize matrices

Answer 11

1. Find the eigenvalues
2. Find n linearly indendent eigenvectors.
3. Construct P from the eigenvectors.
4. Construct D from the corresponding eigenvalues.