南京航空航天大学Matrix-Theory双语矩阵论期末考试2015

NUAA
3 (the vector space of real polynomials of degree less than 3) defined by
(())'()''()p x xp x p x σ=+.
(1) Find the matrix A representing σ with respect to the ordered basis [21,,x x ] for 3P .
(2) Find a basis for 3P such that with respect to this basis, the matrix B representing σ is diagonal. (3) Find the kernel 〔核〕 and range 〔值域〕of this transformation. Solution: (1)
221022x x x x σσσ===+（）（）（） 002010002A ⎛⎫
⎪
= ⎪ ⎪
⎝⎭
----------------------------------------------------------------------------------------------------------------- (2)
101010001T ⎛⎫ ⎪
= ⎪ ⎪⎝⎭
(The column vectors of T are the eigenvectors of A)
The corresponding eigenvectors in 3P are 1000010002T AT -⎛⎫
⎪
= ⎪ ⎪⎝⎭
(T diagonalizes A ) 22
[1,,1][1,,]x x x x T += . With respect to this new basis
2
[1,,1]x x +, the representing matrix of σis diagonal.
------------------------------------------------------------------------------------------------------------------- (3) The kernel is the subspace consisting of all constant polynomials. The range is the subspace spanned by the vectors 2,1x x +
-----------------------------------------------------------------------------------------------------------------------
Let 110020012A -⎛⎫
⎪
= ⎪ ⎪-⎝⎭
.
(1) Find all determinant divisors and elementary divisors of A .
(2) Find a Jordan canonical form of A .
(3) Compute At e . (Give the details of your computations.) Solution: (1)
1
10020012I A λλλλ-⎛⎫ ⎪-=- ⎪
⎪-⎝⎭
,(特征多项式 2
()(1)(2)p λλλ=--. Eigenvalues are 1, 2, 2.) Determinant divisor of order 1()1D λ=, 2()1D λ=, 23()()(1)(2)D p λλλλ==-- Elementary divisors are 2(1) and (2)λλ-- .
---------------------------------------------------------------------------------------------------------------------- (2) The Jordan canonical form is
100021002J ⎛⎫ ⎪
= ⎪ ⎪⎝⎭
--------------------------------------------------------------------------------------------------------------------------
(3) For eigenvalue 1, 0
100
10011I A ⎛⎫
⎪
-=- ⎪ ⎪-⎝⎭ ， An eigenvector is 1(1,0,0)T p = For eigenvalue 2, 11020
00010I A ⎛⎫
⎪
-= ⎪ ⎪⎝
⎭
, An eigenvector is 2(0,0,1)T p = Solve 32(2)A I p p -=, 331100(2)00000101A I p p --⎛⎫⎛⎫
⎪ ⎪
-== ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭
we obtain that
3(1,1,0)T p =-
101001010P ⎛⎫ ⎪
=- ⎪ ⎪⎝⎭, 111000
1010P -⎛⎫
⎪
= ⎪ ⎪-⎝
⎭ 1At J e Pe P -=222101001100010
00101000010t
t t t e e te e ⎛⎫⎛⎫⎛⎫
⎪ ⎪ ⎪=- ⎪ ⎪ ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝
⎭22220000t t t t t t e e e e te e ⎛⎫
-
⎪= ⎪ ⎪-⎝⎭
--------------------------------------------------------------------------------------------------------------------
Suppose that ∈R A and O I A A =--65.
(1) What are the possible minimal polynomials of A ? Explain.
(2) In each case of part (1), what are the possible characteristic polynomials of A ? Explain.
Solution:
(1) An annihilating polynomial of A is 2
56x x --.
The minimal polynomial of A divides any annihilating polynomial of A. The possible minimal polynomials are
6x -, 1x +, and 256x x --.
---------------------------------------------------------------------------------------------------------------
(2) The minimal polynomial of A divides the characteristic polynomial of A. Since A is a matrix of order 3, the characteristic polynomial of A is of degree 3. The minimal polynomial of A and the characteristic polynomial of A have the same linear factors.
Case 6x -, the characteristic polynomial is 3(6)x - Case 1x +, the characteristic polynomial is 3(1)x + Case 256x x --, the characteristic polynomial is 2(1)(6)x x +- or 2(6)(1)x x -+ -------------------------------------------------------------------------------------------------------------------
Let 120000A ⎛⎫
= ⎪⎝⎭
. Find the Moore-Penrose inverse A +
of A .
Solution:
()12011200000A PG ⎛⎫⎛⎫
=== ⎪ ⎪⎝⎭⎝⎭
1()(1,0)T T P P P P +-==, 111()250T T G G GG +
-⎛⎫
⎪
== ⎪ ⎪⎝⎭
110112(1,0)2
05500
0A G P +++⎛⎫⎛⎫ ⎪
⎪=== ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭
也可以用SVD 求.
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Part II (选做题, 每题10分)
请在以下题目中〔第6至第9题〕选择三题解答. 如果你做了四题，请在题号上画圈标明需要批改的三题. 否那么，阅卷者会随意挑选三题批改，这可能影响你的成绩.
Let 4P be the vector space consisting of all real polynomials of degree less than 4 with
usual addition and scalar multiplication. Let 123,,x x x be three distinct real numbers. For each pair of polynomials f and g in 4P , define 3
1,()()i i i f g f x g x =<>=∑.
Determine whether ,f g <> defines an inner product on 4P or not. Explain.
Let n n A ⨯∈R . Show that if x x A =)(σis the orthogonal projection from n R to )(A R , then
A is symmetric and the eigenvalues of A are all 1’s and 0’s.
Let n n A ⨯∈C . Show that x x A H is real-valued for all n C x ∈if and only if A is Hermitian. Let n n B A ⨯∈C ， be Hermitian matrices, and A be positive definite. Show that AB is
similar to BA , and is similar to a real diagonal matrix.
假设正面不够书写，请写在反面.
123()()()x x x x x x ---. Then ,0f f <>=. But 0f ≠. This does not define an inner product.
For any x , ()()x x T A R A N A ⊥-∈=, ()x x 0T A A -=. Hence, T T A A A =. Thus. T A A =.
From above, we have 2A A =. This will imply that λλ-2is an annihilating polynomial of A. The eigenvalue of A must be the roots of 02=-λλ. Thus, the eigenvalues of A are 1’s and 0’s.
也可以用其它方法.
Since A is nonsingular, 1()AB A BA A -=. Hence, A is similar to BA
Since A is positive definite, there is a nonsingular hermitian matrix P such that H A PP =. 1()H H AB PP B P P BP P -==
Since H P BP is Hermitian, it is similar to a real diagonal matrix.
is similar to H AB P BP , H P BP is similar to a real diagonal matrix. Thus AB is similar to a real diagonal matrix.。