5.2 The Characteristic Equation

recap of properties of determinants

if we’re finding eigenvalues of a matrix, A, they must follow the below requirements

the charactristic equation

det(A-λI)=0

if a is nxn then det(A-λI) is a polynomial of degree n called the ? of A

characteristic polynomial

algebraic multiplicity of an eigenvalue is its multiplicity as a ? of the characteristic polynomial

root

imt continued;

A is invertible if and only if the number ? is not an eigenvalue of A

0

matrix A is similar to B if there is an invertible nxn matrix P such that

B = P^-1AP or A=PBP^-1

if matrices A and B are similar, then they have the same __ __, and hence the same ? with the same algebraic multiplicities

characteristic polynomial, e-values

(T/F) If 0 is an eigenvalue of A, then A is invertible.

false

(T/F) The zero vector is in the eigenspace of A associated with an eigenvalue λ.

True

(T/F) The matrix A and its transpose, AT, have different sets of eigenvalues.

false

(T/F) The matrices A and B−1AB have the same sets of eigenvalues for every invertible matrix B

true

(T/F) If 2 is an eigenvalue of A, then A−2I is not invertible.

true

(T/F) If two matrices have the same set of eigenvalues, then they are similar.

false

(T/F) If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

false

(T/F) The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.

true

T/F) The eigenvalue of the n×n identity matrix is 1 with algebraic multiplicity n.

true

(T/F) The matrix A can have more than n eigenvalues.

false