eigenvectors-eigenvalues.html (3754B)
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>eigenvectors-eigenvalues</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Eigenvectors & eigenvalues"><h2 id="Eigenvectors & eigenvalues">Eigenvectors & eigenvalues</h2></div> 12 <p> 13 let A be n × n, \(x \in \Re^n\) is an eigenvector of A if x ≠ 0 and \(\exists \lambda \in \Re\) such that \(Ax = \lambda x\) 14 </p> 15 16 <p> 17 x is eigenvector with corresponding eigenvalue λ. 18 </p> 19 20 <p> 21 Is a given vector \(u \in \Re^n\) an eigenvector of a given A (n × n)? 22 </p> 23 <ul> 24 <li> 25 Do \(Au\), check if result is a multiple of u. 26 27 </ul> 28 29 <p> 30 Is a given λ an eigenvalue of A? 31 </p> 32 <ul> 33 <li> 34 \(\exists x \ne 0\) such that \(Ax - \lambda x = 0 \leftrightarrow (A-\lambda I_n)x = 0\) with nontrivial solutions. 35 36 </ul> 37 38 <p> 39 The solution set of \((A-\lambda I_n)x = 0\) is the eigenspace corresponding to λ. 40 </p> 41 42 <p> 43 How to find a basis for the eigenspace of a given λ? 44 </p> 45 <ol> 46 <li> 47 calculate matrix for \(A-\lambda I_n\) where n is the number of rows or columns of A 48 49 <li> 50 reduce matrix to reduced echelon form 51 52 <li> 53 express solutions in parametric form (basic variables in terms of free variables) 54 55 <li> 56 basis for eigenspace is the set of the coefficients 57 58 </ol> 59 60 <p> 61 If λ = 0, then Ax = 0 has a nontrivial solution (and A is <em>not</em> invertible). 62 </p> 63 64 <p> 65 Eigenvectors corresponding to distinct eigenvalues are linearly independent. 66 </p> 67 68 <div id="Eigenvectors & eigenvalues-Determinant"><h3 id="Determinant">Determinant</h3></div> 69 <p> 70 Geometric interpretation: let \(A = [a_1 \; a_2]\). then the determinant (absolute value) is the surface area (or volume in 3D): 71 </p> 72 73 <p> 74 <img src="img/determinant-geometric-diagram.png" alt="Determinant geometric diagram" /> 75 </p> 76 77 <p> 78 Let A (n × n). A ~ U without scaling and using <em>r</em> row interchanges. then \(\det A = (-1)^r u_{11} \times \dots \times u_{nn}\) 79 </p> 80 81 <p> 82 A is invertible iff \(\det A \ne 0\) 83 </p> 84 85 <p> 86 \(\det AB = (\det A)(\det B)\) 87 </p> 88 89 <p> 90 λ is an eigenvalue of A iff \(\det (A-\lambda I) = 0\) (the characteristic equation of A) 91 </p> 92 93 <p> 94 The eigenvalues of A (n × n) are the solutions for λ. Multiplicity is the number of solutions for λ. 95 </p> 96 97 <div id="Eigenvectors & eigenvalues-Similarity"><h3 id="Similarity">Similarity</h3></div> 98 <p> 99 given A and B (n × n), A is similar to B if ∃p s.t. \(A = PBP^{-1}\) 100 </p> 101 102 <p> 103 If A and B are similar, then they have the same characteristic polynomials (and the same eigenvalues with the same multiplicities) 104 </p> 105 106 <div id="Eigenvectors & eigenvalues-Diagonalization"><h3 id="Diagonalization">Diagonalization</h3></div> 107 <p> 108 A is diagonalizable if A is similar to a diagonal matrix. 109 </p> 110 111 <p> 112 Diagonalization Theorem: A (n × n) is diagonalizable iff A has n linearly independent eigenvectors (the eigenbasis for \(\Re^n\)) 113 </p> 114 115 <p> 116 \(A = P D P^{-1} \leftrightarrow\) columns of P are linearly independent eigenvectors, and the diagonal values of D are the eigenvalues corresponding to the eigenvectors in P. 117 </p> 118 119 <p> 120 How to diagonalize a matrix: 121 </p> 122 <ol> 123 <li> 124 Find eigenvalues of A 125 126 <li> 127 Find n = λ linearly independent eigenvectors 128 129 <li> 130 Construct \(P = \begin{bmatrix} p_1 & p_2 & \ldots & p_n \end{bmatrix}\) 131 132 <li> 133 Construct D from the corresponding eigenvalues on the diagonal. Order of eigenvalues must match the order for columns of P. 134 135 <li> 136 Check \(A = p D p^{-1} \leftrightarrow Ap = pD\) (if p is invertible) 137 138 </ol> 139 140 <p> 141 If A (n × n) has n distinct eigenvalues, it is diagonalizable. 142 </p> 143 144 </body> 145 </html>