MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture 4-Eigenvalues 特征值 Ref: matlab-Mathematics-Matrices and Linear algebra Solving linear Systems of equations ● Vocabulary: coefficients系数 characteristic polynomial特征多项式 根 characteristic roots特征根 eigenvalues特征值 eigenvector特征向量 identity matrix单位阵 decomposition分解 defective matrix亏损矩阵(即不可对角化矩阵) orthogonal matrix正交阵 unitary matrIx酉阵 nonsymmetrIc非对称的 symmetrIc对称的 roundoff error舍入误差 diagonalize对角化 conjugate transpose共轭转置 associate matrⅸx共轭转置矩阵 defective matrix亏损矩阵(即不可对角化矩阵 not diagonalizable) singular value decomposition奇异值分解 Schur decomposition Schur分解 similar matrⅸx相似矩阵 geometric multiplicity几何重数 algebraic multiplicity代数重数 ● Some functions Eigenvalue &*singular value decomposition ☆ Eigenvalues An eigenvalue and eigenvector of a square matrix A are a scalar 1 and a nonzero vector v that satisfy Av=11 Eigenvalue Decomposition With the eigenvalues on the diagonal of a diagonal matrix A and the corresponding tors forming the columns of a matrix v we hay AV=vA If is nonsingular, this becomes the eigenvalue decomposition A=VAV-I a good example is provided by the coefficient matrix of the ordinary differential equation 0-6 > lambda=eig(A) produces a column vector containing the eigenval lambda= Lec4-I
MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec41 MATLAB Lecture 4 – Eigenvalues 特征值 Ref: MATLAB→Mathematics→Matrices and Linear Algebra →Solving Linear Systems of Equations l Vocabulary: coefficients 系数 characteristic polynomial 特征多项式 root 根 characteristic roots 特征根 eigenvalues 特征值 eigenvector 特征向量 identity matrix 单位阵 decomposition 分解 defective matrix 亏损矩阵（即不可对角化矩阵） orthogonal matrix 正交阵 unitary matrix 酉阵 nonsymmetric 非对称的 symmetric 对称的 roundoff error 舍入误差 diagonalize 对角化 conjugate transpose 共轭转置 associate matrix 共轭转置矩阵 defective matrix 亏损矩阵（即不可对角化矩阵 not diagonalizable） singular value decomposition 奇异值分解 Schur decomposition Schur 分解 similar matrix 相似矩阵 geometric multiplicity 几何重数 algebraic multiplicity 代数重数 l Some functions eig poly roots schur svd l Eigenvalue & * singular value decomposition ² Eigenvalues An eigenvalue and eigenvector of a square matrix A are a scalar l and a nonzero vector v that satisfy Av = lv Eigenvalue Decomposition With the eigenvalues on the diagonal of a diagonal matrix L and the corresponding eigenvectors forming the columns of a matrix V, we have AV =VL If V is nonsingular, this becomes the eigenvalue decomposition 1 A V V - = L A good example is provided by the coefficient matrix of the ordinary differential equation >> A =[ 0 6 1 6 2 16 5 20 10]; >> lambda = eig(A) % produces a column vector containing the eigenvalues. lambda =
ATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr 3.0710 24645+176008i 24645-176008 >>VD=eig(A) %computes the eigenvectors and stores the eigenvalues in a diagonal matrix -0.8326 0.2003-0.139410.2003+0.1394i -0.3553 -0.2110-0.64471-0.2110+0.6447i 0.4248 0.6930 -0.6930 -3.0710 0 0 -24645+176008i The first eigenvector is real and the other two vectors are complex conjugates of each other All three vectors are normalized to have Euclidean length, norm(v, 2), equal to one The matrix V" inv(V), which can be written more succinctly as V"D/V, is within roundof error of A. And, inv(V)*A*V, or Va*V, is within roundoff error of D Defective matrices Some matrices do not have an eigenvector decomposition. These matrices are defective,or not diagonalizable. For exa A=[61219 >>VD=eig(A) -0.4741-04082-0.4082 0.81270.81650.8165 -0.3386-04082-0.4082 1.0000 There is a double eigenvalue at 2=1. The second and third columns of v are the same. For this matrix, a full set of linearly independent eigenvectors does not exist The optional Symbolic Math Toolbox extends the capabilities of matlab by connecting to Maple, a powerful computer algebra system. One of the functions provided by the toolbox computes the Jordan Canonical Form. This is appropriate for matrices like our example which is 3-by-3 and has exactly known, integer elements XJI 1.75001.50002.7500 3.0000-3.0000-3.0000 1.25001.5000 Lec4
MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec42 3.0710 2.4645+17.6008i 2.464517.6008i >> [V,D] = eig(A) % computes the eigenvectors and stores the eigenvalues in a … diagonal matrix. V = 0.8326 0.2003 0.1394i 0.2003 + 0.1394i 0.3553 0.2110 0.6447i 0.2110 + 0.6447i 0.4248 0.6930 0.6930 D = 3.0710 0 0 0 2.4645+17.6008i 0 0 0 2.464517.6008i The first eigenvector is real and the other two vectors are complex conjugates of each other. All three vectors are normalized to have Euclidean length, norm(v,2), equal to one. The matrix V*D*inv(V), which can be written more succinctly as V*D/V, is within roundoff error of A. And, inv(V)*A*V, or V\A*V, is within roundoff error of D. Defective Matrices Some matrices do not have an eigenvector decomposition. These matrices are defective, or not diagonalizable. For example, >> A = [ 6 12 19 9 20 33 4 9 15 ]; >> [V,D] = eig(A) V = 0.4741 0.4082 0.4082 0.8127 0.8165 0.8165 0.3386 0.4082 0.4082 D = 1.0000 0 0 0 1.0000 0 0 0 1.0000 There is a double eigenvalue at l = 1. The second and third columns of V are the same. For this matrix, a full set of linearly independent eigenvectors does not exist. The optional Symbolic Math Toolbox extends the capabilities of MATLAB by connecting to Maple, a powerful computer algebra system. One of the functions provided by the toolbox computes the Jordan Canonical Form. This is appropriate for matrices like our example, which is 3by3 and has exactly known, integer elements. >> [X,J] = jordan(A) X = 1.7500 1.5000 2.7500 3.0000 3.0000 3.0000 1.2500 1.5000 1.2500
ATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr 00 0 0 0 The Jordan Canonical Form is an important theoretical concept, but It is not a reliable computational tool for larger matrices, or for matrices whose elements are subject to roundoff errors and other uncertainties Schur Decomposition in MATLAB Matrix Computations The MATLAB advanced matrix computations do not require eigenvalue decompositions They are based, instead, on the Schur decomposition A=USUT where U is an orthogonal matrix and S is a block upper triangular matrix with l-by-l and 2-by-2 blocks on the diagonal. The eigenvalues are revealed by the diagonal elements and blocks of S, while the columns of U provide a basis with much better numerical properties than a set of eigenvectors. The Schur decomposition of our defective example is [U,S]=schur(A) -0.47410.664805774 0.81270.07820.5774 -0.3386-0.74300.57 1000020.784644.6948 1.0000 he double eigenvalue is contained in the lower 2-by-2 block of s Y *Singular Value Decomposition A singular value and corresponding singular vectors of a rectangular matrix A are a scalar o and a pair of vectors u and v that satisfy Av=ou a'u=ov With the singular values on the diagonal of a diagonal matrix 2 and the corresponding singular vectors forming the columns of two orthogonal matrices U and V, we have 4V=U∑.AU=V Since U and v are orthogonal, this becomes the singular value decomposition A=U∑ The full singular value decomposition of an m-by-n matrix involves an m-by-m U, an m-by-n 2, and an n-by-n V. In other words, U and v are both square and 2 is the same size as A. If A has many more rows than columns, the resulting U can be quite large, but most of its columns are multiplied by zeros in 2. In this situation, the economy sized decomposition saves both time and storage by producing an m-by-n U, an n- by-n 2 and the same E Lec4
MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec43 J = 1 0 0 0 1 1 0 0 1 The Jordan Canonical Form is an important theoretical concept, but it is not a reliable computational tool for larger matrices, or for matrices whose elements are subject to roundoff errors and other uncertainties. Schur Decomposition in MATLAB Matrix Computations The MATLAB advanced matrix computations do not require eigenvalue decompositions. They are based, instead, on the Schur decomposition A = USU ' where U is an orthogonal matrix and S is a block upper triangular matrix with 1by1 and 2by2 blocks on the diagonal. The eigenvalues are revealed by the diagonal elements and blocks of S, while the columns of U provide a basis with much better numerical properties than a set of eigenvectors. The Schur decomposition of our defective example is >> [U,S] = schur(A) U = 0.4741 0.6648 0.5774 0.8127 0.0782 0.5774 0.3386 0.7430 0.5774 S = 1.0000 20.7846 44.6948 0 1.0000 0.6096 0 0 1.0000 The double eigenvalue is contained in the lower 2by2 block of S. ² *Singular Value Decomposition A singular value and corresponding singular vectors of a rectangular matrix A are a scalar s and a pair of vectors u and v that satisfy Av = su, A'u =sv With the singular values on the diagonal of a diagonal matrix S and the corresponding singular vectors forming the columns of two orthogonal matrices U and V, we have AV =US, A'U =VS Since U and V are orthogonal, this becomes the singular value decomposition A = USV ' The full singular value decomposition of an mbyn matrix involves an mbym U, an mbyn S , and an nbyn V. In other words, U and V are both square and S is the same size as A. If A has many more rows than columns, the resulting U can be quite large, but most of its columns are multiplied by zeros inS . In this situation, the economy sized decomposition saves both time and storage by producing an mbyn U, an nbyn S and the same V
ATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. On the other hand. the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real nonsymmetric matrix might be complex >>A=9 7 >>US, V=svd(A) %the full singular value decomposition -0.61050.71740.3355 -0.66460.2336-0.7098 -0.43080.656306194 14.9359 05.1883 -0.69250.7214 -0.7214-0.6925 You can verify that U Sy is equal to a to within roundoff error. >>U,S, V=svd(A, 0) %the economy size decomposition is only slightly smaller -0.61050.7174 -0.6646-0.2336 -04308-0.6563 14.9359 05.1883 -0.69250.7214 -0.7214-0.6925 Again, USW is equal to a to within roundoff error ☆ MATLAB >>A=round(10*randn(3) Lec4-4
MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec44 The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. On the other hand, the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category. If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real, nonsymmetric matrix might be complex. >> A =[9 4 6 8 2 7]; >> [U,S,V] = svd(A) %the full singular value decomposition U = 0.6105 0.7174 0.3355 0.6646 0.2336 0.7098 0.4308 0.6563 0.6194 S = 14.9359 0 0 5.1883 0 0 V = 0.6925 0.7214 0.7214 0.6925 You can verify that U*S*V' is equal to A to within roundoff error. >> [U,S,V] = svd(A,0) %the economy size decomposition is only slightly smaller. U = 0.6105 0.7174 0.6646 0.2336 0.4308 0.6563 S = 14.9359 0 0 5.1883 V = 0.6925 0.7214 0.7214 0.6925 Again, U*S*V' is equal to A to within roundoff error. ² MATLAB >> A = round(10*randn(3))
ATLAB Lecture 4 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr A 312 l10 p a= poly (A) computes the coefficients of the characteristic polynomial ofA 1.0e+003* 0.00100.01200.03802.0310 >>roots(p A) 168781 2.4391+10.6951 2.4391-10.695li We usually use eig to compute the eigenvalues of a matrix directly Lec4-5
MATLAB Lecture 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec45 A = 4 3 12 17 11 0 1 12 3 >> p_A = poly(A) % computes the coefficients of the characteristic polynomial of A ans = 1.0e+003 * 0.0010 0.0120 0.0380 2.0310 >> r=roots(p_A) r = 16.8781 2.4391 +10.6951i 2.4391 10.6951i We usually use eig to compute the eigenvalues of a matrix directly