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第1页 （共6页） Matrix Theory, Final, Test Date： 2015年12月28日 矩阵论班号: 学号 姓名 必做题（70分） 题号 得分 1 2 3 4 5 选做题（30分） 总分 Part I (必做题，共5题，70分) 第1题（15分） 得分 Let P[?1,1] denote the set of all real polynomials of degree less than 3 with domain（定义域） [?1,1]. The addition and scalar multiplication are defined in the usual way. Define an inner product on P[?1,1] by ?p,q???p(t)q(t)dt. ?11(1) Construct an orthonormal basis for P[?1,1] from the basis 1,x,x2 by using the Gram-Schmidt orthogonalization process. (2) Letf(x)?x2?1?P[?1,1]. Find the projection of f onto the subspace spanned by{1,x}. Solution: (1) 1??1,1?? p1??x,121?1?1dx?2, u1?12, ?121?[?112?1xdx]12?0, u2?x?p1?x?p13x, 2 x2?p210331?(3x2?1) p2??x,???x,x?x?, u3?24223x?p22222 ------------------------------------------------------------------------------------------- (2) proj??x2?1,u1?u1??x2?1,u2?u2??x2?1, ??12?12??x2?1,33x?x 222212?0?? 332 ---------------------------------------------------------------------------------------------------------------- 精品文档
第2题（15分） 得分 Let ?be the linear transformation on P3 (the vector space of real polynomials of degree less than 3) defined by
?(p(x))?xp'(x)?p''(x).
(1) Find the matrix A representing ? with respect to the ordered basis [1,x,x2] for P3.
(2) Find a basis for P3 such that with respect to this basis, the matrix B representing ? is diagonal.
(3) Find the kernel（核） and range （值域）of this transformation. Solution: (1)
?（）1?0?002??010 ? A??（x）?x??
?002??（x2）?2?2x2??----------------------------------------------------------------------------------------------------------------- (2)
?101???T??010? (The column vectors of T are the eigenvectors of A)
?001???The corresponding eigenvectors in P3 are ?000??010 T?1AT???? (T diagonalizes A) ?002???2 [1,x,x2?1]?[1,x,x2]T . With respect to this new basis [1,x,x?1], the representing
?matrix of is diagonal.
------------------------------------------------------------------------------------------------------------------- (3) The kernel is the subspace consisting of all constant polynomials.
The range is the subspace spanned by the vectors x,x2?1
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第3题（20分） 得分 ?1?10??Let A???020?.
?0?12???(1) Find all determinant divisors and elementary divisors ofA.
(2) Find a Jordan canonical form of A.
(3) Compute eAt. (Give the details of your computations.) Solution: (1)
10????1??I?A????20?,(特征多项式 p(?)?(??1)(??2)2. Eigenvalues are 1, 2, ?0?01??2???2.)
Determinant divisor of order D1(?)?1, D2(?)?1, D3(?)?p(?)?(??1)(??2)2 Elementary divisors are (??1) and (??2)2 .
---------------------------------------------------------------------------------------------------------------------- (2) The Jordan canonical form is
?100??? J??021?
?002???--------------------------------------------------------------------------------------------------------------------------
?0? (3) For eigenvalue 1, I?A??0?0??1? For eigenvalue 2, 2I?A??0?0?1?110??0? ， An eigenvector is p1?(1,0,0)T ?1??0??0?, An eigenvector is p2?(0,0,1)T 0??。
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??1?10??0????Solve (A?2I)p3?p2, (A?2I)p3???000?p3??0?we obtain that
?0?10??1????? p3?(1,?1,0)T
?101??11?? P??00?1?, P?1???00?010??0?1???t?101??e??? eAt?PeJP?1??00?1??0?010??0???0??1? 0??00??110??etet?e2t0??????e2t0? e2tte2t??001???02t???e2t?0e2t????0?10??0?te--------------------------------------------------------------------------------------------------------------------
第4题（10分） 得分 Suppose that A?R3?3 and A2?5A?6I?O.
(1) What are the possible minimal polynomials ofA? Explain.
(2) In each case of part (1), what are the possible characteristic polynomials ofA? Explain.
Solution:
(1) An annihilating polynomial of A is x?5x?6.
The minimal polynomial of A divides any annihilating polynomial of A. The possible minimal polynomials are
x?6, x?1, and x2?5x?6.
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(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 x?6, the characteristic polynomial is (x?6)3 Case x?1, the characteristic polynomial is (x?1)3 Case x2?5x?6, the characteristic polynomial is (x?1)2(x?6) or (x?6)2(x?1)
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第5题（10分） 得分 Let A???120???. Find the Moore-Penrose inverse A of A.
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Solution: A??
P??(PTP)?1PT?(1,0), G??1??111???A??G?P???2?(1,0)??25??5?0???0?1?1???GT(GGT)?1??2?
5???0?0??0? 0???120??1??????120??PG 000???0?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. 否则，阅卷者会随意挑选三题批改，这可能影响你的成绩.
第6题 Let P4 be the vector space consisting of all real polynomials of degree less
than 4 with usual addition and scalar multiplication. Let x1,x2,x3 be three distinct real numbers. For each pair of polynomials f and g inP4, define ?f,g???f(xi)g(xi).
i?13Determine whether ?f,g? defines an inner product on P4 or not. Explain.
第7题 LetA?Rn?n. Show that if ?(x)?Axis the orthogonal projection from
RntoR(A), then A is symmetric and the eigenvalues of A are all 1’s and 0’s.
第8题 LetA?Cn?n. Show thatxHAxis real-valued for allx?Cnif and only if A is Hermitian. 第9题 Let A，B?Cn?n be Hermitian matrices, and A be positive definite. Show that
AB is similar to BA , and is similar to a real diagonal matrix.
若正面不够书写，请写在反面. 选做题得分
第6题解答 Let f(x)?(x?x1)(x?x2)(x?x3). Then ?f,f??0. But f?0. This does not define an inner product. 第7题解答 For any x, Ax?x ?R(A)??N(AT), AT(Ax?x) ?0. Hence, ATA?AT. Thus. A?AT.
From above, we have A2?A. This will imply that ?2??is an annihilating polynomial of A. The eigenvalue of A must be the roots of ?2???0. Thus, the eigenvalues of A are
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南京航空航天大学Matrix-Theory双语矩阵论期末考试2015
