Let A E Cnxn be skew-Hermitian (i.e., A* = -A). (a) State the spectral theorem for...

 

Transcribed Image Text:

Let A E Cnxn be skew-Hermitian (i.e., A* = -A). (a) State the spectral theorem for normal matrices. (b) Show that all the eigenvalues of A are purely imaginary (i.e., their real parts are zero). (c) Argue that A-I is nonsingular and prove that B = (A-1)-¹(A+I) is unitary. A [9 (d) Let X = matrix norm. (ii) What is the spectral radius of X* for k≥ 1? Justify your answer. (iii) Find lim Σ Α'. 100 [2] [3] E Cmxm with m > n and assume that ||A|| < 1, where || || is any subordinate (i) Show that S = span{en+1,...,em) is an invariant subspace for X, where e, denotes the jth column of the mxm identity matrix, and that X has at least m-n eigenvalues equal to 1. [3] [4] [5] -00 X*. Hint: Since ||A|| < 1 you may use without proof that (I-A)-¹ [3] Let A E Cnxn be skew-Hermitian (i.e., A* = -A). (a) State the spectral theorem for normal matrices. (b) Show that all the eigenvalues of A are purely imaginary (i.e., their real parts are zero). (c) Argue that A-I is nonsingular and prove that B = (A-1)-¹(A+I) is unitary. A [9 (d) Let X = matrix norm. (ii) What is the spectral radius of X* for k≥ 1? Justify your answer. (iii) Find lim Σ Α'. 100 [2] [3] E Cmxm with m > n and assume that ||A|| < 1, where || || is any subordinate (i) Show that S = span{en+1,...,em) is an invariant subspace for X, where e, denotes the jth column of the mxm identity matrix, and that X has at least m-n eigenvalues equal to 1. [3] [4] [5] -00 X*. Hint: Since ||A|| < 1 you may use without proof that (I-A)-¹ [3]