厦门大学数学科学学院：《高等代数》课程教学资源（应用与实验）MATLAB Ex 4 - Eigenvalue

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¢)