Math Problem Statement
Find the SVD
Find the SVD of the matrix: [[1, -1], [0, 1], [1, 0]]
Solution
You have provided a matrix image, which seems to be the following matrix :
1 & -1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$$ To compute the Singular Value Decomposition (SVD) of a matrix $$A$$, we seek to express $$A$$ in the form: $$A = U \Sigma V^T$$ where: - $$U$$ is a $$3 \times 3$$ orthogonal matrix, - $$\Sigma$$ is a $$3 \times 2$$ diagonal matrix with singular values on the diagonal, - $$V^T$$ is the transpose of a $$2 \times 2$$ orthogonal matrix $$V$$. ### Steps to Find the SVD: 1. **Compute $$A^T A$$:** $$A^T A = \begin{pmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$$ 2. **Find Eigenvalues of $$A^T A$$:** To find the singular values, we solve for the eigenvalues of $$A^T A$$. The characteristic equation is: $$\det(A^T A - \lambda I) = 0$$ $$\det \begin{pmatrix} 2-\lambda & -1 \\ -1 & 2-\lambda \end{pmatrix} = 0$$ Solving this gives: $$(2-\lambda)^2 - (-1)(-1) = 0 \implies \lambda^2 - 4\lambda + 3 = 0$$ $$\lambda = 3 \quad \text{and} \quad \lambda = 1$$ The square roots of these eigenvalues are the singular values: $$\sigma_1 = \sqrt{3}, \quad \sigma_2 = 1$$ 3. **Find the right singular vectors (columns of $$V$$):** Solve $$(A^T A - \lambda I)v = 0$$ for each eigenvalue $$\lambda$$. For $$\lambda = 3$$, solve: $$\begin{pmatrix} -1 & -1 \\ -1 & -1 \end{pmatrix} v = 0$$ This yields $$v_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}$$. For $$\lambda = 1$$, solve: $$\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} v = 0$$ This yields $$v_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$. Hence, $$V = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$. 4. **Compute $$U$$ using $$Av = u\sigma$$:** Now, use the formula $$u = \frac{Av}{\sigma}$$ to find the left singular vectors $$u_1$$ and $$u_2$$. Would you like further details or clarifications on any part? Let me know! ### Related Questions: 1. How are the eigenvalues of $$A^T A$$ connected to singular values? 2. Can you always decompose any matrix into an SVD? 3. What is the significance of the matrices $$U$$, $$\Sigma$$, and $$V$$ in applications? 4. How is the SVD related to the rank of a matrix? 5. How do we apply SVD in image compression? **Tip:** The columns of $$U$$ are the left singular vectors of $$A$$, and they correspond to the directions in which $$A$$ stretches vectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Singular Value Decomposition
Eigenvalues and Eigenvectors
Formulas
SVD: A = U Σ V^T
Eigenvalue equation: det(A^T A - λI) = 0
Singular values: σ = sqrt(λ)
Theorems
Spectral Theorem
SVD Theorem
Suitable Grade Level
University-level Linear Algebra
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