矩阵论_研究生期末考试_2016_安丰稳

(1) Suppose A Prove eA (2) Conversely, suppose eA Is it true that A = holds? (Hint. Consider the distance from a point to a subspace.) 6. (10%) Let B be a complex matrix of order n. For each eigenvalue 2 C of the matrix B; put E ( ) = fz 2 Cn : B z = z g: (1) Let B be a normal matrix and let 1 ; 2 ; ; s 2 C be all the distinct eigenvalues of B . Is it true that there is an orthogonal sum Cn = E ( 1 ) E ( 2) E ( s) 3, =e : =e : = :
(1) Find the Jordan form of A. (2) Evaluate the matrix
A2017 + A2016 + A: 2. (20%) Let B be a real symmetric matrix of order 3. For each column vector x 2 R3 , de…ne a function f (x) : R3 ! R; x 7 ! f (x) = x| B x: (1) Prove that B is a positive matrix () f (x) reaches its minimal value 0 at only one point in R3 . (2) Let 1; 1; 3 be all the eigenvalues of B: Find the set of values of the function f (x) on the unit sphere of R3 , i.e., ff (x) 2 R : x 2 R3 with kxk = 1g: 1
Wuhan University 2016-2017 School Year, The 1st Semester A …nal examination of Theory of Matrices for the …rst year graduate students
Please note that here you can use any results proved in class or in the textbook; but you should indicate clearly what you are using. Thank you. 1. (20%) Given a matrix of order 3 0 1 0 0 1 A = @ 1 0 0 A: 0 1 0
(2) Suppose the matrix B has n distinct eigenvalues. Take a complex matrix T of order n. Prove B T =T B () There are complex numbers k0 ; k1 ; T = k0 I + k1 B + ; kr源自文库; ; kn + kn
(1) Find the matrix Af which represents the linear operator f (relative to the standard linear basis of C3 ). (2) Find the norm of the linear operator f . Here C3 is given with the standard inner product. (3) Find the adjoint operator f of the linear operator f . (Hint. Via the conjugate transpose of the matrix Af .) (4) Find a positive constant c 2 R such that kAf z k holds for each column vector z 2 C3 . 4. (20%) Let T be a complex matrix of order n. For the C-linear operator T : Cn ! Cn ; x 7 ! T x consider the two subspaces ker (T ) = fx 2 Cn : T x = 0g; (1) Evaluate (2) Evaluate dimC img (T ) rank (T ) : (3) Prove that the linear operator T : Cn ! Cn is injective () The linear operator T : Cn ! Cn is surjective. (4) Suppose T is a hermitian matrix. Prove there is an orthogonal sum ker (T ) img (T ) = Cn : img (T ) = fT x 2 Cn : x 2 Cn g: dimC ker (T ) + rank (T ) : c kz k
3. (20%) Let f : C3 ! C3 be a C-linear operator given by 0 1 0 1 z1 z1 z2 z = @ z2 A 2 C3 7 ! f (z ) = @ z1 + z3 A 2 C3 : z3 z2 + z3
5. (10%) Let A be a complex matrix of order n. Let eA be the complex matrix of order n given by the exponent function ex on C. Take a complex number 2 C and a complex column vector 2 Cn . 2
for the complex vector space Cn with the standard inner product ? Here, for mutually orthogonal subspaces V1 ; V2 ; ; Vs of Cn with s we de…ne V1 V2 V3 = (V1 V2 ) V3 ; V1 in the same way, V1 V2 Vs = (V1 V2 Vs 1 ) Vs : V2 V3 V4 = (V1 V2 V3 ) V4 ;
1 1
2 C such that Bn 1:
+ kr B r +
Here I denotes the identity matrix of order n. 3