symmetric-matrices.html (2363B)
1 <!DOCTYPE html> 2 <html> 3 <head> 4 <script type="text/javascript" async src="https://cdn.jsdelivr.net/gh/mathjax/MathJax@2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> 5 <link rel="Stylesheet" type="text/css" href="style.css"> 6 <title>symmetric-matrices</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Symmetric matrices"><h2 id="Symmetric matrices">Symmetric matrices</h2></div> 12 <p> 13 symmetric if \(A^T = A\) (also has to be square) 14 </p> 15 16 <div id="Symmetric matrices-Diagonalization of symmetric matrices"><h3 id="Diagonalization of symmetric matrices">Diagonalization of symmetric matrices</h3></div> 17 <p> 18 If A is symmetric, any two eigenvectors from two different eigenspaces are orthogonal. 19 </p> 20 21 <p> 22 An n×n matrix is orthogonally diagonalizable iff A is symmetric. 23 </p> 24 25 <p> 26 An n×n matrix A: 27 </p> 28 <ul> 29 <li> 30 has n real eigenvalues, including multiplicities 31 32 <li> 33 is orthogonally diagonalizable 34 35 <li> 36 dimension of eigenspace for each eigenvalue λ == multiplicity of λ as root of the characteristic equation (\(\det (A-\lambda I) = 0\)) 37 38 <li> 39 eigenspaces are mutually orthogonal (i.e. eigenvectors corresponding to different eigenvalues are orthogonal) 40 41 </ul> 42 43 <div id="Symmetric matrices-Singular value decomposition"><h3 id="Singular value decomposition">Singular value decomposition</h3></div> 44 <p> 45 singular values: square roots of eigenvalues of \(A^T A\), denoted by \(\sigma_1, \dots, \sigma_n\) in ascending order. They are also the lengths of vectors \(Av_1, \dots, Av_n\). 46 </p> 47 48 <p> 49 Suppose \(\{v_1, \dots, v_n\}\) is an orthonormal basis for \(\Re^n\) consisting of eigenvectors of \(A^T A\) in ascending order, and suppose A has r nonzero singular values. 50 </p> 51 <ul> 52 <li> 53 Then \(\{Av_1, \dots, Av_n\}\) is orthogonal basis for Col A, and rank == r. 54 55 <li> 56 Then there exists Σ matrix m×n for which diagonal entries are first r singular values of A, and there exist matrices U (orthogonal, m²) and V (orthogonal, n²) such that \(A = U \Sigma V T\). 57 58 <li> 59 Col U are "left singular vectors" of A, Col V are "right singular vectors" of A. 60 61 </ul> 62 63 <p> 64 Let A be n², then the fact that "A is invertible" means that: 65 </p> 66 <ul> 67 <li> 68 \((\text{Col} A)^\perp = \{ 0 \}\) 69 70 <li> 71 \((\text{Nul} A)^\perp = \Re^n\) 72 73 <li> 74 \(\text{Row} A = \Re^n\) 75 76 <li> 77 A has n nonzero singular values 78 79 </ul> 80 81 </body> 82 </html>