Math Problem Statement

how to Find the singular values and the SVD

Solution

To find the singular values and compute the Singular Value Decomposition (SVD) of a matrix AA, follow these steps:

1. Definition of SVD

The Singular Value Decomposition of a matrix AA is expressed as: A=UΣVTA = U \Sigma V^T where:

  • UU: Orthogonal matrix containing the left singular vectors of AA.
  • Σ\Sigma: Diagonal matrix with singular values of AA.
  • VV: Orthogonal matrix containing the right singular vectors of AA.

2. Steps to Compute SVD

  1. Find ATAA^T A and AATA A^T: Compute the symmetric matrices ATAA^T A and AATA A^T.

  2. Compute Eigenvalues and Eigenvectors:

    • Find the eigenvalues (λ\lambda) and eigenvectors of ATAA^T A.
    • Similarly, find the eigenvalues (λ\lambda) and eigenvectors of AATA A^T.
  3. Determine Singular Values:

    • The singular values are the square roots of the eigenvalues of ATAA^T A (or AATA A^T).
    • Arrange them in descending order: σ1σ20\sigma_1 \geq \sigma_2 \geq \dots \geq 0.
  4. Construct UU, Σ\Sigma, and VV:

    • VV: Form VV using the eigenvectors of ATAA^T A.
    • Σ\Sigma: Place the singular values σ1,σ2,\sigma_1, \sigma_2, \dots on the diagonal matrix Σ\Sigma (non-diagonal entries are zero).
    • UU: Form UU using the eigenvectors of AATA A^T.
  5. Verify the Decomposition: Multiply UΣVTU \Sigma V^T to confirm it equals AA.

3. Example

Let A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.

  1. Compute ATAA^T A and AATA A^T: ATA=[10141420],AAT=[5111125].A^T A = \begin{bmatrix} 10 & 14 \\ 14 & 20 \end{bmatrix}, \quad A A^T = \begin{bmatrix} 5 & 11 \\ 11 & 25 \end{bmatrix}.

  2. Find eigenvalues:

    • Eigenvalues of ATAA^T A: λ1=29.44\lambda_1 = 29.44, λ2=0.56\lambda_2 = 0.56.
  3. Compute singular values:

    • σ1=λ1=5.42\sigma_1 = \sqrt{\lambda_1} = 5.42, σ2=λ2=0.75\sigma_2 = \sqrt{\lambda_2} = 0.75.
  4. Form UU, Σ\Sigma, VV:

    • Use eigenvectors to construct UU and VV.
    • Diagonalize Σ\Sigma with σ1\sigma_1 and σ2\sigma_2.

Tip

If the matrix is large or computation-intensive, software like Python (numpy.linalg.svd), MATLAB (svd), or any numerical tool can quickly compute the SVD.


Would you like to solve an example step-by-step or learn the application of SVD in practical contexts?

Related Questions:

  1. What are some real-world applications of SVD?
  2. How does the SVD differ from Eigen Decomposition?
  3. How can you compute SVD using Python or MATLAB?
  4. What is the significance of the singular values in data compression?
  5. What are the computational complexities of calculating SVD?

Tip: Always check the condition number of a matrix (σmax/σmin\sigma_{\max} / \sigma_{\min}) to determine its numerical stability.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Decomposition
Singular Value Decomposition (SVD)

Formulas

A = U Σ V^T
Singular values: σ_i = sqrt(λ_i), where λ_i are eigenvalues of A^T A

Theorems

Spectral Theorem (used to find eigenvalues and eigenvectors of symmetric matrices)
SVD Theorem (every matrix has a singular value decomposition)

Suitable Grade Level

Undergraduate (Linear Algebra)