MATLAB Exercise School of mathematical Sciences xiamen univers http:/edjpkc.xmu.ed MATLAB Exercise 4-Eigenvalue 1. Find the eigenvalues and the corresponding eigenspaces for each of the following matrices And judge whether it is a defective matrix 001 23 001 05-1 01-1 Help Select Matlab Help in the toolbar, then select Index and input eig, distinguish the difference usage of this function. For example [v, D]=eig(A) and [ v, d]=eig(A, B) 2. Judge whether the following two matrix are similar? 200 200 A=040,B 102 -362 3. Use poly and roots function to compute the characteristic polynomial and characteristic roots of a random 4 4 matrix. According to the result show that the characteristic polynomial and characteristic roots of a in mathematics formula 4. In each of the following, factor the matrix A into a product XDX, where D is diagonal.(use two different methods) 56 A 2)A= 3)A 100 221 4)A=-213 5)A=012 6)A=24-2 5. For each of the following, find a matrix B such thatB=A 5 00 0 6. Given A= 1 2 0 find an orthogonal matrix U that diagonalizes A Please display the results in rational format. (scht Ex4-1
MATLAB Exercise 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Ex41 MATLAB Exercise 4 – Eigenvalue 1. Find the eigenvalues and the corresponding eigenspaces for each of the following matrices. And judge whether it is a defective matrix. 1) 3 2 3 2 Ê ˆ Á ˜ Ë - ¯ 2) 6 4 3 1 Ê - ˆ Á ˜ Ë - ¯ 3) 3 1 1 1 Ê - ˆ Á ˜ Ë ¯ 4) 3 8 2 3 Ê - ˆ Á ˜ Ë ¯ 5) 1 1 2 3 Ê ˆ Á ˜ Ë- ¯ 6) 0 1 0 0 0 1 0 0 1 Ê ˆ Á ˜ Á ˜ Á ˜ Ë ¯ 7) 1 1 1 0 2 1 0 0 1 Ê ˆ Á ˜ Á ˜ Á ˜ Ë ¯ 8) 1 1 1 0 3 1 0 5 1 Ê ˆ Á ˜ Á ˜ Á ˜ - Ë ¯ 9) 4 5 1 1 0 1 0 1 1 Ê - ˆ Á ˜ - Á ˜ Á ˜ - Ë ¯ Help Select Matlab Help in the toolbar, then select Index and input eig, distinguish the difference usage of this function. For example [V,D] = eig(A) and [V,D] = eig(A,B). 2. Judge whether the following two matrix are similar? 2 0 0 2 0 0 0 4 0 , 1 4 0 1 0 2 3 6 2 A B Ê ˆ Ê ˆ Á ˜ Á ˜ = = - Á ˜ Á ˜ Á ˜ Á ˜ - Ë ¯ Ë ¯ 3. Use poly and roots function to compute the characteristic polynomial and characteristic roots of a random 4×4 matrix. According to the result show that the characteristic polynomial and characteristic roots of A in mathematics formula. 4. In each of the following, factor the matrix A into a product XDX 1 , where D is diagonal. (use two different methods) 1) 0 1 1 0 A Ê ˆ = Á ˜ Ë ¯ 2) 5 6 2 2 A Ê ˆ = Á ˜ Ë- - ¯ 3) 2 8 1 4 A Ê - ˆ = Á ˜ Ë - ¯ 4) 1 0 0 2 1 3 1 1 1 A Ê ˆ Á ˜ = - Á ˜ Á ˜ - Ë ¯ 5) 2 2 1 0 1 2 0 0 1 A Ê ˆ Á ˜ = Á ˜ Á ˜ - Ë ¯ 6) 1 2 1 2 4 2 3 6 3 A Ê - ˆ Á ˜ = - Á ˜ Á ˜ - Ë ¯ 5. For each of the following, find a matrix B such that B2 = A. 1) 2 1 2 1 A Ê ˆ = Á ˜ Ë- - ¯ 2) 9 5 3 0 4 3 0 0 1 A Ê - ˆ Á ˜ = Á ˜ Á ˜ Ë ¯ 6. Given 0 1 1 1 2 0 1 0 3 A Ê - ˆ Á ˜ = Á ˜ Á ˜ - Ë ¯ , find an orthogonal matrix U that diagonalizes A. Please display the results in rational format. (schur)
MATLAB Exercise 4 School of mathematical Sciences xiamen univers http:/edjpkc.xmu.ed Help Select Matlab Help in the toolbar, then select Index and input schur, distinguish the difference usage of this function. For example T=schur(A); [U, T=schur(A) 7. Let A=2 3. Compute the singular values and the singular value decomposition of A Compare square of the singular values of A with the eigenvalues of A'A. Are they the same? Help Select Matlab Help in the toolbar, then select Index and input svd to know the usage of his function 8. Let A=-17 -11 0. Compute the eigenvalues of A by roots and eig functions 1123 respectively. Compute its eigenvalue decomposition, Schur decomposition and singular value decomposition respectively. Compare the results and show the differences 9. Please generate 4 symmetric matrices and check the following proposition "If the eigenvalues of a symmetric matrix A are A,1,.,A,then the singular values of A areA1l,|A2l…,n|” 10. Please generate 10 matrices( some of them are singular奇异, others are nonsingular非奇异 Bp A]ig, others are diagonalized matrices and others are not diagonalized matrices)and calculate their eigenvalues and singular values Then count the number of nonzero eigenvalues and singular values Show the conclusion you guess 11. *Compute the singular value decomposition of matrix 6 335 2 1) Use the singular value decomposition to find orthonormal bases for R(A )and N(a) 2)Use the singular value decomposition to find orthonormal bases for R(A) and N(A) Ex4-2
MATLAB Exercise 4 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Ex42 Help Select Matlab Help in the toolbar, then select Index and input schur, distinguish the difference usage of this function. For example T = schur(A); [U,T] = schur(A). 7. Let 1 1 2 3 1 0 A Ê ˆ Á ˜ = Á ˜ Á ˜ Ë ¯ . Compute the singular values and the singular value decomposition of A. Compare square of the singular values of A with the eigenvalues of A’A. Are they the same? Help Select Matlab Help in the toolbar, then select Index and input svd to know the usage of this function. 8. Let 4 3 12 17 11 0 1 12 3 A Ê - ˆ Á ˜ = - - Á ˜ Á ˜ Ë ¯ . Compute the eigenvalues of A by roots and eig functions respectively. Compute its eigenvalue decomposition, Schur decomposition and singular value decomposition respectively. Compare the results and show the differences. 9. Please generate 4 symmetric matrices and check the following proposition “If the eigenvalues of a symmetric matrix A are 1 2 , ,..., l l ln , then the singular values of A are 1 2 | |,| |,...,| | l l ln ”. 10. Please generate 10 matrices (some of them are singular 奇异, others are nonsingular 非奇异 即可逆, others are diagonalized matrices and others are not diagonalized matrices) and calculate their eigenvalues and singular values. Then count the number of nonzero eigenvalues and singular values. Show the conclusion you guess. 11. *Compute the singular value decomposition of matrix 2 5 4 6 3 0 6 3 0 2 5 4 A Ê ˆ Á ˜ Á ˜ = Á ˜ Á ˜ Ë ¯ . 1) Use the singular value decomposition to find orthonormal bases for R(A') and N(A) 2) Use the singular value decomposition to find orthonormal bases for R(A) and N(A¢)
