2. Matrix B is defined as, -31 B=10-2 0 0-2 a. Find the Jordan form of matrix B b. Find exp (Jt), where J is the Jordan form of matrix B c. Find exp (Bt) 2. Matrix B is defined as, -31 B=10-2...
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 2 1 0 (a) Find a Jordan form J for A. (b) Find the change of basis matrix X such that X AX -J Problem. Let A=1-1-2-2-2 0-2 1 1 -1 2 1 0 (a) Find a Jordan form J for A. (b) Find the change of basis matrix X such that X AX -J
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J. Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J.
Exercise 30. Let A be a 5 x 5 matrix. Find the Jordan canonical form J under each of the following assumptions (i) A has only eigenvalue namely 4 and dim N(A- 41) = 4. one (ii) dim N(A 21) = 5. (ii dim N(A -I) = 3 and dim N (A 31) 2. (iv) det(A I) = (1 - )2(2 - A)2 (3 - ) and dim N(A - I) dim N(A - 21) 1 (v) A5 0 and...
(14) (7 marks) Find a Jordan canonical form of the matrix 13 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3) 0 0
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
Problem 1 (a) If z = exp(bt), where a and b are not functions of t, then dz/dt =? (b) 2 = sin (Tī ) cos (TÊ), then the ABSOLUTE VALUE of 'Lo zdx =? Problem 2 (a) What is the most general solution to the following differential equa
for a matrix solution of the quadratic (3) Find a formula of the form x = -B C equation ax2 + bx +c = 0. Here c denotes and 0 denotes 0 0 (Hint: First show how the square root of any number D can be obtained using a where it looks different depending matrix of the form on whether D is negative. Then use the quadratic formula.) positive or for a matrix solution of the quadratic (3) Find a...
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
Matrix Algebra Given a system of the form -M1X1 + X2 = bt -m2X1 + X2 = b 2 where my, m2, 61, and b2 are constants, (a) Show that the system will have a unique solution if my # m2. (b) Show that if m1 = m2, then the system will be consistent only if b1 = b2. (c) Give a geometric interpretation of parts (a) and (b).