Chapter 5: Eigenvalues and Eigenvectors

Question 1

Characteristic equation of A

Answer 1

det(λI - A) = 0

Question 2

Theorem 5.1.1 Eigenvalue & characteristic equation

Answer 2

λ is an eigenvalue of A iff:

det(λI - A) = 0

Question 3

Theorem 5.1.2:

Eigenvalues of an nxn triangular matrix:

Answer 3

Are the entries on the main diagonal.

Question 4

Theorem 5.1.4 on powers of A and λ

Answer 4

If k is a positive integer, λ is an eigenvalue of matrix A, and x is a corresponding eigenvector, then λ^k is an eigenvalue of A^k and x is a corresponding eigenvector.

Question 5

Theorem 5.1.5 on invertibility of square matrices and λ

Answer 5

A square matrix A is invertible iff λ = 0 is not an eigenvalue of A.

Question 6

Similar matrices

Answer 6

If A and B are square matrices, then we say that B is similar to A if there is an invertible matrix P such that B = P^-1AP.

Question 7

Diagonalisable matrix

Answer 7

A square matrix A is said to be diagonalisable if it is similar to some diagonal matrix.
If there exists some invertible matrix P such that P^-1AP is diagonal.
P is said to diagonalise A.

Question 8

Theorem 5.2.1:

Equivalent statements on diagonalizability and eigenvectors.

Answer 8

If A is an n x n matrix:

• A is diagonalisable
• A has n linearly independent eigenvectors

Question 9

Procedure for Diagonalizing a Matrix

Answer 9

• Confirm that the matrix is actually diagonalizable by finding n linearly independent eigenvectors. One way to do this is by finding a basis for each eigenspace and merging these basis vectors into a single set S. If this set has fewer than n vectors, then the matrix is not diagonalizable.
• Form the matrix P = [p1 p2 … pn] that has the vectors in S as its column vectors.
• The matrix P^-1AP will be the diagonal and have the eigenvalues λ1, λ2, … λn corresponding to the eigenvectors p1, p2, …, pn as its successive diagonal entries.

Question 10

Theorem 5.2.2 on eigenvectors and linear independence

Answer 10

If v1, v2, …, vk are eigenvectors of a matrix corresponding to distinct eigenvalues, then {v1, v2, …, vk} is a linearly independent set.

Question 11

Theorem 5.2.3 on diagonalisability and distinct eigenvectors

Answer 11

If an n x n matrix has n distinct eigenvalues, then A is diagonalizable.

Question 12

Similarity invariant between A a

Answer 12

A