南航双语矩阵论期中考试卷mid-term-exam(2014)优选全文

Mid-term Exam of Matrix Theory (2014)
Preferentially Selected Five Questions (5×20 )
Q1.Given A ∈P n ×n ,consider the following questions.
1)If A is invertible,prove that A −1can be represented by the polynomial of A with degree less than n .
2)For any positive integer k ∈N ,prove that A k can be represented by the polynomial of A with degree less than n .3)Especially A = 11221
,find the representative polynomials of A −1and A 2014as men-tioned in 1)and 2).
Q2.Denote A a linear transformation in R 3,α1,α2,α3the basis of R 3.Suppose that the representation matrix of A with respect to α1,α2,α3is A = 120
20−2−2−1
.1)Show that β1=α1,β2=α1+α2,β1=α1+α2+α3also form a basis of R 3.
2)Determine the representative matrix of A with respect to β1,β2,β2.
3)Find the eigenvalues and eigenvectors of A .
Q3.Denote R [x ]3to be the vector space of zero and polynomials with degree less than 3.
1)Determine the dimension of R [x ]3and give a basis of R [x ]3.
2)Define the linear transformation D on R [x ]3,
D (f (x ))=f (x ),
∀f (x )∈R [x ]3.Show R (D )and ker(D ).
3)Prove that D is not diagonalizable.
4)Define the inner product on R [x ]3,
(f,g )= 1
−1f (x )g (x )dx,
∀f (x ),g (x )∈R [x ]3,
please Gram-Schmidt orthogonalize the basis given in 1).
Q4.1)To the best of your knowledge about λ−matrix,determine if the following two matrices are similar or not,and give reason,
1
A =
210021002 ,B = 2a 0
02a 002 .
2)Denote V ={ a 11a 12a 21a 22
∈R 2×2|a 11=a 22}.i)Find a basis of V and show the dimension.ii)Arbitrarily given A = a 11a 12a 21a 22 and B = b 11b 12b 21b 22
in V ,define (A,B )=a 11b 11+2a 12b 12+a 21b 21.
Please show that (A,B )is an inner product on V .
Q5.Given A ∈C m ×n and b ∈C m ,please prove
1)there exists a real number α>0such that A H A +αI is nonsingular;
2)the solution to the least square problem min x ∈C n
{ Ax −b 2+α x 2}is x ∗=(A H A +αI )−1A H b ,where · stands for the 2−norm in C m .Q6.Given A ∈R n ×n ,summarize the necessary and sufficient conditions of A to be di-agonalizable,and prove at least one of them.Determine if the matrix A given in Q2is diagonalizable or not.If yes,please explain why,if not,please give the Jordan canonical form of A .
2。