CS475 / CM375 Lecture 14: Oct 27, 2011 Eigenvalue problems Reading: - - PDF document

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CS475 / CM375 Lecture 14: Oct 27, 2011 Eigenvalue problems Reading: - - PDF document

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26/10/2011 CS475 / CM375 Lecture 14: Oct 27, 2011 Eigenvalue problems Reading: [TB] Chapters 24, 25 CS475/CM375 (c) 2011 P. Poupart & J. Wan 1 Eigenvalue problem 1 0 0 Example: 2 1 2 4 4 5 det

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CS475 / CM375 Lecture 14: Oct 27, 2011

Eigenvalue problems Reading: [TB] Chapters 24, 25

CS475/CM375 (c) 2011 P. Poupart & J. Wan 1

Eigenvalue problem

  • Example:

1 2 1 2 4 4 5 det

CS475/CM375 (c) 2011 P. Poupart & J. Wan 2

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Example continued

  • 1:
  • 3:

CS475/CM375 (c) 2011 P. Poupart & J. Wan 3

Example Continued

  • Thus

Λ

  • Note: we never compute eigenvalues by finding the

roots of the characteristic polynomial

CS475/CM375 (c) 2011 P. Poupart & J. Wan 4

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Eigenvalues

  • Gershgorin Theorem: Let be any square matrix. The

eigenvalues of are located in the union of the disks: ∑

  • Proof: Consider , such that , 0.

Scale such that

1

for some Then ∑

  • ∑
  • ⟹ ∑
  • ∑
  • ∑

||

  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

5

Example

4 0.5 0.6 5 0.6 0.5 3

  • Picture:

CS475/CM375 (c) 2011 P. Poupart & J. Wan 6

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Rayleigh quotient

  • Assume is real and symmetric. Thus has

real eigenvalues and a complete set of

  • rthogonal eigenvectors

, … , , , … ,

  • 1
  • Def: The Rayleigh quotient of a vector is:
  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

7

Rayleigh quotient

  • Notes

1. If is an eigenvector, then is an eigenvalue 2. Given , find such that min

  • ⋮
  • ( 1 least squares)

The normal equations:

  • 3. Theorem: Let be an eigenvector and

then

  • as →

CS475/CM375 (c) 2011 P. Poupart & J. Wan 8

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Power Iteration

  • Let approximate eigenvector,

1 and set of eigenvectors

  • Then ⋯

⟹ ⋯

CS475/CM375 (c) 2011 P. Poupart & J. Wan 9

Power Iteration

  • Similarly,

⋯

  • ⋯
  • Suppose ⋯ ||.

Then

  • → 0 as → ∞

~

  • for large

i.e., ~

CS475/CM375 (c) 2011 P. Poupart & J. Wan 10

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Example

21 7 1 5 7 7 4 4 20 1 1 1

CS475/CM375 (c) 2011 P. Poupart & J. Wan 11

Power Iteration Algorithm

initial guess, 1 for 1,2, …

  • Rayleigh quotient

end

CS475/CM375 (c) 2011 P. Poupart & J. Wan 12

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Power Iteration Algorithm

  • Notes

1. We normalize in each computation of 2. Theorem: Suppose ⋯ and

  • 0. Then
  • and
  • as → ∞

3. It only computes 4. The convergence is linear, the convergence rate

  • 5.

The convergence can be slow if

CS475/CM375 (c) 2011 P. Poupart & J. Wan 13

Inverse Iteration

  • Idea 1: Use to compute the smallest eigenvalue

Note: Λ

  • ,
  • , … ,
  • Thus: ⋯
  • ⋯
  • ⋮
  • ⋯
  • ⋯
  • ∴
  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

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Inverse Iteration

  • Idea 2: Shifting. Consider

∉ Λ Then has the same eigenvectors as and its eigenvalues are , where ∈ Λ.

  • If is close to , then would be the smallest

eigenvalue of .

  • We can apply idea 1 to compute

CS475/CM375 (c) 2011 P. Poupart & J. Wan 15

Example

21 7 1 5 7 7 4 4 20 Λ 8,16,24 , 15

CS475/CM375 (c) 2011 P. Poupart & J. Wan 16