12. Let C be a consumption matrix such that Cm →0 as m-o, and for m1,2,.... let Dm C++ C. Find a ...
Let matrix M = -8 -24 -12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP^−1. If not, explain carefully why not.
Let matrix M = -8 -24 12 0 4 0 6 12 10 (a) Find the eigenvalues of M (b) For each eigenvalue λ of M, find a basis for the eigenspace of λ. (c) Is the matrix M diagonalizable? If so, find matrices D and P such that D is a diagonal matrix and M=PDP−1. If not, explain carefully why not.
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
Find the inverse, if it exists, of the given matrix 1 0 0 OA. 0 1 1 0 0 1 1 0 0 2-1 1 Find the inverse, if it exists, of the given matrix. 5 12 5 2 A. 12 5 5 -12 -2 5 -5 2 12 -5 -5-12 -25 OB. O c. O D. Determine whether the two matrices are inverses of each other by computing their product. 9 4-22 2 -45 O No O Yes
)-( 1 (c) Let C be a real 3 x 3 matrix and b be a real 3-vector. The general solution to the matrix equation Cx=b is given by 2 2 =X3 + -4 2 for all XER Let 10 y = -6 8 (i) Let z be a real 3-vector. Find the solution set to the matrix equation Cz=0 (ii) Calculate M1, M2 ER such that 2 y = M1 ( 3 + H2 ·()--() 1 (iii) Express Cy...
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0. (b) Show that there is no 5 x 5 matrix B such that B2 + 15 = 0. (c) Let C be any n xn matrix such that C2 + In 0. Let l be any eigenvalue of C. Show that 12 Conclude that C has no real eigenvalues. [1] [3] =-1. [3]
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
Answer
part c (ii)
(b) Let 12 -1 2 0 -3 0 om = A -5/ Compute the spectral radius of A.- a system of linear equations (c) Suppose a certain iterative scheme used to solve a system of lines is an invertih an invertible matrix Ax = b is given by QxK+1 = (Q - A) bu oxK+1 = (0 - AX" + b, where Prove that (1) (ii) exll s ||1 - Q - A||llex-1|| lexll s ||1...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent matrix and use that along with the power series definition of the matrix exponential to find eAt 2!
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent...