Math Problem Statement
Construct the Singular Value Decomposition for the matrix A = [[2, 2, -2], [2, 2, -2], [-2, -2, 6]].
Solution
The task is to perform the Singular Value Decomposition (SVD) for the matrix , which is:
2 & 2 & -2 \\ 2 & 2 & -2 \\ -2 & -2 & 6 \end{bmatrix}$$ ### Steps for SVD: Singular Value Decomposition expresses any matrix $$A$$ as the product of three matrices: $$A = U \Sigma V^T$$ where: - $$U$$ is an orthogonal matrix (left singular vectors), - $$\Sigma$$ is a diagonal matrix with non-negative real numbers (singular values), - $$V^T$$ is the transpose of an orthogonal matrix (right singular vectors). Let’s compute the SVD of the given matrix $$A$$. I'll begin by calculating the eigenvalues of $$A^T A$$ and proceed to compute the singular values and vectors.The **Singular Value Decomposition (SVD)** of the matrix $$A$$ is: $$A = U \Sigma V^T$$ Where: 1. The left singular matrix $$U$$ is: $$U = \begin{bmatrix} -0.4082 & -0.5774 & 0.7071 \\ -0.4082 & -0.5774 & -0.7071 \\ 0.8165 & -0.5774 & 0.0000 \end{bmatrix}$$ 2. The diagonal matrix $$\Sigma$$ (singular values) is: $$\Sigma = \begin{bmatrix} 8 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ 3. The right singular matrix $$V^T$$ is: $$V^T = \begin{bmatrix} -0.4082 & -0.4082 & 0.8165 \\ -0.5774 & -0.5774 & -0.5774 \\ -0.7071 & 0.7071 & 0.0000 \end{bmatrix}$$ Thus, the matrix $$A$$ can be decomposed into these three matrices. Do you want further details about how SVD works or its applications? ### Related Questions: 1. What is the geometric interpretation of singular values and vectors? 2. How are the singular values related to the rank of the matrix? 3. What is the difference between SVD and Eigenvalue Decomposition? 4. Can SVD be used for solving systems of linear equations? 5. How does truncating the SVD help in data compression (e.g., in image compression)? ### Tip: SVD is widely used in principal component analysis (PCA) to reduce the dimensionality of data while preserving as much variance as possible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Decomposition
Formulas
SVD: A = UΣV^T
Eigenvalue decomposition for computing V and Σ
Theorems
SVD theorem
Suitable Grade Level
University Level
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