5

5.1

© 2012 Pearson Education, Inc.

Eigenvalues and Eigenvectors

EIGENVECTORS AND

EIGENVALUES

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EIGENVECTORS AND EIGENVALUES

Definition: An eigenvector of an nxn matrix A is a

nonzero vector x such that Ax=λx for some 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.

Eigenvectors may NOT be 0

Eigenvalue may be 0

Eigenvectors - Example

Are u and v eigenvectors of A, for:

𝐴 =1 65 2

, 𝐮 =6−5

, 𝑣 =3−2

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Finding Eigenvectors - Example

Show that 7 is an eigenvalue of A from

previous example and find corresponding

eigenvectors.

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Eigenspace

General solution of Ax=λx is same as Nul

space of (A – λI)

Subspace of Rn

Called “Eigenspace”

Contains 0 and all eigenvectors

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Finding Basis for Eigenspace - Example

A has eigenvalue of 2. Find basis for

corresponding eigenspace.

𝐴 =4 −1 62 1 62 −1 8

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Geometric Interpretation

The eigenspace, shown in the following figure, is

a two-dimensional subspace of R3.

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EIGENVALUES of Triangular Matrix

Theorem 1: The eigenvalues of a triangular

matrix are the entries on its main diagonal.

Proof: For simplicity, consider the 3x3 case.

If A is upper triangular, the (A – λI) has the form

11 12 13

22 23

33

11 12 13

22 23

33

λ 0 0

λ 0 0 λ 0

0 0 0 0 λ

λ

0 λ

0 0 λ

a a a

A I a a

a

a a a

a a

a

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EIGENVALUES of Triangular Matrix

The scalar λ is an eigenvalue of A if and only if the equation (A – λI) = 0 has a nontrivial solution, i.e., iff the equation has a free variable.

Because of the zero entries in (A – λI), it is easy to see that (A – λI) = 0 has a free variable if and only if at least one of the entries on the diagonal of A – λI is zero.

This happens if and only if λ equals one of the entries a11, a22, a33 in A.

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EIGENVECTORS – Linear Independence

Theorem 2: If v1, …, vr are eigenvectors that

correspond to distinct eigenvalues λ1, …, λr of an

nxn matrix A, then the set {v1, …, vr} is linearly

independent.

0 as Eigenvalue

If 0 is an eigenvalue:

Ax=0x=0 has non-trivial solution

A is NOT invertible

Additions to IMT:

(s) 0 is not an eigenvalue of A

(t) |A| ≠ 0

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5

5.1

© 2012 Pearson Education, Inc.

Eigenvalues and Eigenvectors

THE CHARACTERISTIC

EQUATION

Finding Eigenvalues - Example

Find the eigenvalues of 𝐴 =2 33 −6

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Characteristic Equation

characteristic equation: |A – λI| = 0

Scalar equation

Degree n, where A is nxn

Solutions are eigenvalues of A

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THE CHARACTERISTIC EQUATION

Example 2: Find the characteristic equation of

5 2 6 1

0 3 8 0

0 0 5 4

0 0 0 1

A

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SIMILARITY

If A and B are nxn matrices, then A is similar to

B if there is an invertible matrix P such that

P-1AP = B, or, equivalently

A = PBP-1

Changing A into B is called a similarity

transformation.

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SIMILARITY

Theorem 4: If nxn matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).