南京航空航天大学Matrix-Theory双语矩阵论期末考试2015.doc

精品文档 Part I (必做题，共5题，70分) 第1题（15分）得分 Let denote the set of all real polynomials of degree less than 3 with domain（定义域） . The addition and scalar multiplication are defined in the usual way. Define an inner product on by .(1) Construct an orthonormal basis for from the basis by using the Gram-Schmidt orthogonalization process. (2) Let. Find the projection of onto the subspace spanned by.Solution: (1) , , , , , - (2) -第2题（15分）得分 Let be the linear transformation on (the vector space of real polynomials of degree less than 3) defined by . (1) Find the matrix representing with respect to the ordered basis for . (2) Find a basis for 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) - (2) (The column vectors of T are the eigenvectors of A) The corresponding eigenvectors in are (T diagonalizes A) . With respect to this new basis , 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 -第3题（20分）得分Let . (1) Find all determinant divisors and elementary divisors of. (2) Find a Jordan canonical form of . (3) Compute . (Give the details of your computations.) Solution: (1),(特征多项式 . Eigenvalues are 1, 2, 2.)Determinant divisor of order , , Elementary divisors are . - (2) The Jordan canonical form is - (3) For eigenvalue 1, ， An eigenvector is For eigenvalue 2, , An eigenvector is Solve , we obtain that , - 第4题（10分）得分Suppose that and . (1) What are the possible minimal polynomials of? Explain.(2) In each case of part (1), what are the possible characteristic polynomials of? Explain.Solution: (1) An annihilating polynomial of A is . The minimal polynomial of A divides any annihilating polynomial of A. The possible minimal polynomials are, , and . -(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 , the characteristic polynomial is Case , the characteristic polynomial is Case , the characteristic polynomial is or -第5题（10分）得分Let . Find the Moore-Penrose inverse of . Solution: , 也可以用SVD求. -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 be the vector space consisting of all real polynomials of degree less than 4 with usual addition and scalar multiplication. Let be three distinct real numbers. For each pair of polynomials and in, define . Determine whether defines an inner product on or not. Explain. 第7题 Let. Show that if is the orthogonal projection from to, then A is symmetric and the eigenvalues of A are all 1s and 0s. 第8题 Let. Show thatis real-valued for allif and only if A is Hermitian. 第9题 Let be Hermitian matrices, and A be positive definite. Show that AB is similar to BA , and is similar to a real diagonal matrix. 选做题得分若正面不够书写，请写在反面. 第6题解答 Let . Then . But . This does not define an inner product. 第7题解答For any x, , . Hence, . Thus. . From above, we have . This will imply that is an annihilating polynomial of A. The eigenvalue of A must be the roots of . Thus, the eigenvalues of A are 1s and 0s. 第8题解答 See Thm 7.1.1, page 182. 也可以用其它方法. 第9题解答 Since A is nonsingular, . Hence, A is similar to BA Since A is positive definite, there is a nonsingular h