Question

1. If the ax matrix A has eigenvalues ....., what are the eigenvalues of a) 4*, where & is a positive integer. AE? A ' b) ', assuming the inverse matrix exists. c) A' (transpose of ). d) a, where a is a real number. e) Is there any relationship between the eigenvalues of 'A and those of the A matrix? Hint: Use to justify your answer. 2. Compute the spectral norm of 0 0 b) c) c) 1-1 0 1 1+1 3. Given nxn matrix A and an axl column vector x, prove a) Asahi b) | 4 where is any eigenvalue of A c) LAB141181 4. If x0) is an xl differentiable function of. (o is also a differentiable function of 1, and (c) is continuously differentiable while '() Wt, prove that 0 60 *** Note: use Euclidean norm. € (0,0) aceh moun ) ('0) ---((09)<"O vic(-,0) 5. Given the following nonlinear state equation: (1) = f(x,1)= *20-24 (0)2(6) 7-**(6)+x+ (-)+x36)+-(0) a) Determine the equilibrium point(s) of the system

Answer 1

4. al att Given Matrix A has eigen values d.... An Ak suppose A has eigen value & with eigen vector X les AX AX Now Ark= A (AX) = A ČAX) = A (AX) L:14 # scalar] = A(4x) similarly we get A²x= 3 x 44 x = 14x Ak has eigen values at .... and Al - Multiply the ean Ax=AX by N'A APAY (Axle AYAT AX + t'x = Atx (A1A=I ] Thus Al has elegenvalle & Corresponding to elgenvector X . Al hais eigen valies died, Ant

c) Att det CAT - XI) - det (AT-IT) [ IT = Il det (A-MI)T = det (AHI) For any solution IA-111=0 is a solution of - IAT-Aal=0 1 A & AT have same eigen vectors d) aA la Al = QIAN 1 colution t é la A-Allo is solution with multiplier - it ala-di1=0 a A has eigen waren da, atzina An