Chapter 5

Question 1

Eigenvector

Answer 1

A nonzero vector x such that Ax = lambdax (Where A is an nxn matrix) for some scalar lambda.

Question 2

Eigenvalue

Answer 2

A scalar lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax = lambda*x

Question 3

Theorem 1

Answer 3

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

Question 4

Theorem 2

Answer 4

If v1,…,vr are eigenvectors that correspond to distinct eigen values of an nxn matrix then the set {v1,…,vr} is linearly independent

Question 5

Similarity

Answer 5

If A and B are nxn matrices, then A is similar to B if there is an invertible matrix Psuch that P^-1AP = B or A = PBP^-1

Question 6

Theorem 4

Answer 6

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

Question 7

Theorem 5

Answer 7

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

Question 8

Theorem 6

Answer 8

An nxn matrix with n distinct eigenvalues is diagonalizable