Module 7 Linear Algebra: Definitions and Theorems

If n is a positive integer, λ is an eigenvalue of A, then λ n is an eigenvalue of A^n

• True

• False

0 is an eigenvalue of A if and only if A does not have an inverse.

• True

• False

A and A T have the same eigenvalue.

• True

• False

If A is a nxn matrix, then the following sentences are equivalent:

1. A is diagonalizable

2. A has n linearly independent eigenvectors

• True

• False

Anxn is diagonalizable if and only if there are n linearly independent eigenvectors.

• True

• False

If v1 , v2 , …,vk are eigenvectors of A corresponding to distinct eigenvalues λ1 , λ2 , … λk , then { v1 , v2 , …,vk } is a linearly independent set.

• True

• False

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

• True

• False

If A is a square matrix, then:

1. For every eigenvalue of A, its geometric multiplicity is less than or equal to its algebraic multiplicity.

2. A is diagonalizable if and only if the geometric multiplicity is equal to its algebraic multiplicity for each of the eigenvalue of A.

• True

• False

If A is a square matrix, then the following sentences are equivalent:

a. A is orthogonally diagonalizable

b. A has an orthonormal set that consists of n eigenvectors

c. A is symmetric

• True

• False