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Computing Eigenvectors and Eigenvalues

Module by: Richard Baraniuk

Summary: An overview of the concepts behind computing eigenvectors and eigenvalues.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Step 1: Find the Eigenvalues

Find λ λ such that Av=λv A v λ v with v≠0 v 0 .

Av−λv=0 A v λ v 0 (1)
Av−λIv=0 A v λ I v 0 (2)
where I I is the indentity matrix in the form 1000010000......00...1 1 0 0 0 0 1 0 0 0 0 ... ... 0 0 ... 1
A−λIv=0 A λ I v 0 (3)
This means, matrix A−λI A λ I is not invertible or
detA−λI=0 A λ I 0 (4)
where det˙ ˙ is the determinant.

So solve detA−λI=0 A λ I 0 which will be a polynomial of order N N; λ λ's are its roots.

Example 1

A=3-1-13 A 3 -1 -1 3

A−λI=3-1-13−λ00λ=3−λ-1-13−λ A λ I 3 -1 -1 3 λ 0 0 λ 3 λ -1 -1 3 λ (5)
detA−λI=det3−λ-1-13−λ=3−λ2− -1 2=λ2−6λ+8=λ−2λ−4=0 A λ I 3 λ -1 -1 3 λ 3 λ 2 -1 2 λ 2 6 λ 8 λ 2 λ 4 0 (6)
Therefore the eigenvalues are λ 1 =2 λ 1 2 , and λ 2 =4 λ 2 4 .

Step 2: Find the Eigenvectors

For each λ i λ i solve the N×N N N set of linear equations for vi v i

Avi=λvi A v i λ v i (7)
with N N equations in N N unknowns.

Example 2

A=3-1-13 A 3 -1 -1 3 , λ 1 =2 λ 1 2 , λ 2 =4 λ 2 4

for λ 1 =2 λ 1 2 ,

3-1-13v11v12=2v11v12 3 -1 -1 3 v 11 v 12 2 v 11 v 12 (8)
v1=v11v12=11 v 1 v 11 v 12 1 1 (9)

for λ 2 =4 λ 2 4 ,

3-1-13v21v22=4v21v22 3 -1 -1 3 v 21 v 22 4 v 21 v 22 (10)
v2=v21v22=1-1 v 2 v 21 v 22 1 -1 (11)

Question:

What should be done with "repeated eigenvalues" (ie: when λ i = λ j λ i λ j for some λ i ≠ λ j λ i λ j )?

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