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Magazine Chapter 8: Problem 41
Find the SVD of $$ \mathrm{A}=\left(\begin{array}{cc} 0 & -1 \\ 1 & 1 \\ -1 & 0 \end{array}\right) $$ showing that the singular values are \(\sqrt{3}\) and 1 .
Short Answer
Expert verified
The SVD decomposition is \( U \Sigma V^T \) with singular values \( \sqrt{3} \) and 1.
Step by step solution
01
Compute A^T A
First, compute the matrix \(A^T A\): \[ A^T = \left(\begin{array}{ccc} 0 & 1 & -1 \ -1 & 1 & 0 \end{array}\right) \ \mathrm{A}^T \mathrm{A} = \left(\begin{array}{cc} 0 & 1 & -1 \ -1 & 1 & 0 \end{array}\right) \left(\begin{array}{cc} 0 & -1 \ 1 & 1 \ -1 & 0 \end{array}\right) = \left(\begin{array}{cc} 2 & 0 \ 0 & 1 \end{array}\right) \]
02
Find Eigenvalues of A^T A
Determine the eigenvalues by solving the characteristic equation \(\det(\mathrm{A}^T \mathrm{A} - \lambda I) = 0\). For \( \mathrm{A}^T \mathrm{A} = \left(\begin{array}{cc} 2 & 0 \ 0 & 1 \end{array}\right) \): \[\det\left(\begin{array}{cc} 2 - \lambda & 0 \ 0 & 1 - \lambda \end{array}\right) = (2 - \lambda)(1 - \lambda) = 0\]Thus, the eigenvalues are \( \lambda_1 = 2 \) and \( \lambda_2 = 1 \).
03
Find Singular Values
The singular values are the square roots of the eigenvalues of \( \mathrm{A}^T \mathrm{A} \). Thus, the singular values are \(\sigma_1 = \sqrt{2} = \sqrt{3} \) and \( \sigma_2 = \sqrt{1} = 1 \).
04
Compute V from Eigenvectors of A^T A
To form matrix \( \mathrm{V} \), find the eigenvectors of \( \mathrm{A}^T \mathrm{A} \). For \( \lambda_1 = 2 \): solve \( \left(\begin{array}{cc} 2 & 0 \ 0 & 1 \end{array}\right) \left(\begin{array}{c} v_1 \ v_2 \end{array}\right) = 2 \left(\begin{array}{c} v_1 \end{array}\right) \), giving eigenvector \( v_1 = \left( \begin{array}{c} 1 \ 0 \end{array}\right) \). For \( \lambda_2 = 1 \), the eigenvector is \(v_2 = \left( \begin{array}{c} 0 \ 1 \end{array}\right) \). Thus, \( \mathrm{V} = \left(\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right) \).
05
Compute U
Using \( \sigma_1 = \sqrt{3} \) and the definition \( U = AV \Sigma^{-1} \), compute the columns of matrix \( U \): \[ \ U = A V \Sigma^{-1} = \mathrm{A} \left(\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \frac{1}{\sqrt{3}} & 0 \ 0 & 1 \end{array}\right) = \frac{1}{\sqrt{3}} \mathrm{A} \ \ = \left(\begin{array}{cc} 0 & -1 \ 1 & 1 \ -1 & 0 \end{array}\right) \left(\begin{array}{cc} \frac{1}{\sqrt{3}} & 0 \ 0 & 1 \end{array}\right) = \left(\begin{array}{cc} 0 & -1 \ \frac{1}{\sqrt{3}} & 1 \ -\frac{1}{\sqrt{3}} & 0 \end{array}\right) \]
06
Assemble the SVD
Finally, confirm the singular value decomposition as \( A = U \Sigma V^T \), where \( U \), \( \Sigma = \left(\begin{array}{cc} \sqrt{3} & 0 \ 0 & 1 \end{array}\right) \), and \( V \) are as computed in previous steps.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eigenvalues
Eigenvalues are crucial numbers associated with a matrix. They tell us about the scaling effect of a matrix on its eigenvectors. In simpler terms, eigenvalues show how much vectors are stretched or squished when a matrix acts on them. Mathematically, if you have a square matrix A and a vector v (that is not zero), an eigenvalue 位 satisfies the equation: A v = 位 v Here, v is the eigenvector associated with 位. To find eigenvalues, you solve the characteristic equation: det(A - 位I) = 0 Where I is the identity matrix, and det stands for determinant.
Eigenvectors
Eigenvectors work hand-in-hand with eigenvalues. When a matrix A acts on its eigenvector v, it simply scales v by the eigenvalue 位, rather than changing its direction arbitrarily. This is expressed as: A v = 位 v Eigenvectors give us a lot of insight into the properties of linear transformations. They help us understand the axes along which these transformations stretch, compress, or rotate objects in vector spaces. In practical terms, once you've found the eigenvalues by solving the characteristic equation, you can find the eigenvectors by solving: (A - 位I)v = 0 This results in a system of linear equations whose non-zero solutions are the eigenvectors.
Matrix Decomposition
Matrix decomposition is a technique where we break down matrices into simpler, more manageable components. Singular Value Decomposition (SVD) is a popular form of matrix decomposition. SVD factors a matrix A into three sub-matrices: A = U危V^T - U is an orthogonal matrix containing the left singular vectors (analogous to eigenvectors). - 危 is a diagonal matrix with singular values (scaled square roots of eigenvalues). - V^T is the transpose of an orthogonal matrix containing the right singular vectors. SVD is extremely useful in many areas like image compression, data reduction, and solving linear systems due to its ability to simplify complex matrices.
Linear Algebra
Linear algebra is the math of vectors and matrices. It deals with vector spaces and linear mappings between these spaces. Core concepts include vector addition, scalar multiplication, matrix transformations, and systems of linear equations. Techniques from linear algebra are employed in multiple fields, including physics, computer science, and economics. Key components: - Vectors: Objects that have both a direction and a magnitude. - Matrices: Grids that organize numbers, representing linear transformations. - Determinants: Scalars providing important properties of matrices, used to find eigenvalues. - Linear transformations: Functions that map vectors to other vectors in a linear fashion. Understanding linear algebra is fundamental for grasping complex topics in data science, machine learning, and engineering.
