Let A ben x n matrix with eigenvalues A, A,..., An a) Show that the determinant...

 

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Let A ben x n matrix with eigenvalues A, A,..., An a) Show that the determinant of A is equal to the product of its eigenvalues, i.e., det(A) = I=1^j. [20 points] b) The trace of a matrix is defined to be the sum of its diagonal entries, i.e., trace (A) = =1 ajj. Show that the trace of A is equal to the sum of its eigenvalues, i.e., trace(A) = -1;. [20 points] Hint: Consider the characteristic polynomial of A (Slide 12 in Lecture 4). 2. [20 points] Consider A is an n*n matrix. (i) Create your own code for QR factorization of A matrix using Householder method [15 points] (ii) Use your codes in part (i) for QR factorization of the below matrix: 8 10 121 A 17 9 22. [5 points] 13 21 181] 3. [30 points] Implement your own code for inverse iteration with a shift to compute the eigenvalue nearest to 2, and normalized eigenvector, of the matrix: [2 A = 7 4 3 8 6 18 10 12] 4. [30 points] Write your own code for Rayleigh quotient iteration for computing an eigenvalue and corresponding eigenvector of matrix. To test your code, solve the previous example in problem 3. Remark: Recall that you can only use essential built-in commands in MATLAB such as size, length, zeros, eye and for/if statements for all problems. Let A ben x n matrix with eigenvalues A, A,..., An a) Show that the determinant of A is equal to the product of its eigenvalues, i.e., det(A) = I=1^j. [20 points] b) The trace of a matrix is defined to be the sum of its diagonal entries, i.e., trace (A) = =1 ajj. Show that the trace of A is equal to the sum of its eigenvalues, i.e., trace(A) = -1;. [20 points] Hint: Consider the characteristic polynomial of A (Slide 12 in Lecture 4). 2. [20 points] Consider A is an n*n matrix. (i) Create your own code for QR factorization of A matrix using Householder method [15 points] (ii) Use your codes in part (i) for QR factorization of the below matrix: 8 10 121 A 17 9 22. [5 points] 13 21 181] 3. [30 points] Implement your own code for inverse iteration with a shift to compute the eigenvalue nearest to 2, and normalized eigenvector, of the matrix: [2 A = 7 4 3 8 6 18 10 12] 4. [30 points] Write your own code for Rayleigh quotient iteration for computing an eigenvalue and corresponding eigenvector of matrix. To test your code, solve the previous example in problem 3. Remark: Recall that you can only use essential built-in commands in MATLAB such as size, length, zeros, eye and for/if statements for all problems.