Math Problem Statement

Solve

Find the singular value decomposition of the matrix A = [[0, 1], [1, 0], [0, 2], [1, 1], [2, 1]].

Solution

To solve this problem, we need to find the Singular Value Decomposition (SVD) of the given matrix AA.

The matrix AA is:

0 & 1 \\ 1 & 0 \\ 0 & 2 \\ 1 & 1 \\ 2 & 1 \end{pmatrix}$$ The Singular Value Decomposition (SVD) of a matrix $$A$$ is of the form: $$A = U \Sigma V^T$$ Where: - $$U$$ is an orthogonal matrix of size $$m \times m$$ (in this case $$5 \times 5$$). - $$\Sigma$$ is a diagonal matrix (of size $$5 \times 2$$) containing the singular values. - $$V$$ is an orthogonal matrix of size $$2 \times 2$$, and $$V^T$$ is its transpose. ### Steps to compute the SVD: 1. **Find $$A^T A$$**: First, calculate the matrix $$A^T A$$ (where $$A^T$$ is the transpose of $$A$$). $$A^T = \begin{pmatrix} 0 & 1 & 0 & 1 & 2 \\ 1 & 0 & 2 & 1 & 1 \end{pmatrix}$$ $$A^T A = \begin{pmatrix} 0 & 1 & 0 & 1 & 2 \\ 1 & 0 & 2 & 1 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 2 \\ 1 & 1 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ 3 & 7 \end{pmatrix}$$ 2. **Eigenvalues and eigenvectors of $$A^T A$$**: Find the eigenvalues and eigenvectors of $$A^T A$$. The eigenvalues correspond to the squares of the singular values $$\sigma_1^2, \sigma_2^2$$, and the eigenvectors give the matrix $$V$$. For the matrix $$\begin{pmatrix} 6 & 3 \\ 3 & 7 \end{pmatrix}$$, solving the characteristic equation: $$\text{det}(A^T A - \lambda I) = 0$$ $$\begin{vmatrix} 6 - \lambda & 3 \\ 3 & 7 - \lambda \end{vmatrix} = 0$$ Gives the eigenvalues $$\lambda_1 = 9$$ and $$\lambda_2 = 4$$. Thus, the singular values are: $$\sigma_1 = \sqrt{9} = 3, \quad \sigma_2 = \sqrt{4} = 2$$ The corresponding eigenvectors are found to construct $$V$$. 3. **Construct $$V$$ and $$\Sigma$$**: The eigenvectors corresponding to $$\lambda_1 = 9$$ and $$\lambda_2 = 4$$ form the columns of the matrix $$V$$. The diagonal matrix $$\Sigma$$ is: $$\Sigma = \begin{pmatrix} 3 & 0 \\ 0 & 2 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}$$ 4. **Find $$U$$**: The matrix $$U$$ is constructed using the formula: $$U = \frac{1}{\sigma_i} A v_i$$ Where $$v_i$$ are the eigenvectors of $$A^T A$$, and $$\sigma_i$$ are the singular values. Once all matrices are computed, the SVD is complete with $$A = U \Sigma V^T$$. Do you need more details or have any questions? Here are 5 follow-up questions for deeper understanding: 1. What are the properties of the matrices $$U$$, $$\Sigma$$, and $$V$$ in an SVD? 2. How can SVD be used to approximate a matrix by reducing its rank? 3. Why is the matrix $$A^T A$$ always symmetric and positive semi-definite? 4. How are the singular values related to the eigenvalues of $$A^T A$$? 5. In what applications would you use the SVD, such as in data compression or machine learning? **Tip**: When working with SVD, always remember that the singular values in $$\Sigma$$ are non-negative and sorted in descending order.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Singular Value Decomposition (SVD)
Matrix Decomposition

Formulas

Singular Value Decomposition: A = UΣV^T
Eigenvalues and Eigenvectors of A^T A

Theorems

Spectral Theorem for Symmetric Matrices
Properties of Orthogonal Matrices

Suitable Grade Level

College Level