南航双语矩阵论matrix theory第7章部分习题参考答案.doc

第七章部分习题参考答案Exercise 1Show that a normal matrix A is Hermitian if its eigenvalues are all real.Proof If A is a normal matrix, then there is a unitary matrix that diagonalizes A. That is, there is a unitary matrix U such that where D is a diagonal matrix and the diagonal elements of D are eigenvalues of A. If eigenvalues of A are all real, then Therefore, A is Hermitian.Exercise 2Let A and B be Hermitian matrices of the same order. Show that AB is Hermitian if and only if .Proof If , then . Hence, AB is Hermitian. Conversely, if AB is Hermitian, then . Therefore, .Exercise 3 Let A and B be Hermitian matrices of the same order. Show that A and B are similar if they have the same characteristic polynomial. Proof Since matrix A and B have the same characteristic polynomial, they have the same eigenvalues. There exist unitary matrices U and V such that , . Thus, . ()That is . Hence, A and B are similar. Exercise 4 Let A be a skew-Hermitian matrix, i.e., , show that(a) and are invertible.(b) is a unitary matrix with eigenvalues not equal to .Proof of Part (a)Method 1: (a) since , it follows that For any Hence, is positive definite. It follows that is invertible. Hence, both and are invertible.Method 2: If is singular, then there exists a nonzero vector x such that . Thus, ,. (1) Since is real, it follows that. That is . Since , it follows that (2)Equation (1) and (2) implies that . This contradicts the assumption that x is nonzero.Therefore, is invertible. Method 3: Let be an eigenvalue of A and x be an associated eigenvector. . Hence, is either zero or pure imaginary. 1 and can not be eigenvalues of A. Hence, and are invertible.Method 4: Since , A is normal. There exists a unitary matrix U such that Each is pure imaginary or zero. Since for , det. Hence, is invertible. Similarly, we can prove that is invertible. Proof of Part (b) Method 1: Since , it follows that ( Note that if P is nonsingular.)Hence, is a unitary matrix. Denote .Since , Hence, can not be an eigenvalue of .Method 2:By method 4 of the Proof of Part (a), The eigenvalues of are , which are all not equal to .Method 3: Since , it follows that If is an eigenvalue of , then there is a nonzero vector x, such that . That is .It follows that .This implies that . This contradiction shows that can not be an eigenvalue of .Exercise 6 If H is Hermitian, show that is invertible, and is unitary.Proof Let . Then A is skew-Hermitian. By Exercises #4, and are invertible, and is unitary. This finishes the proof.Exercise 7Find the Hermitian matrix for each of the following quadratic forms. And reduce each quadratic form to its canonical form by a unitary transformation (a) Solution , . Eigenvalues of A are , and .Associated unit eigenvectors are , , and , respectively. form an orthonormal set. Let , and . Then we obtain the canonical form Exercise 9Let A and B be Hermitian matrices of order n, and A be positive definite. Show that AB is similar to a real diagonal matrix.Proof Since A is positive definite, there exists an nonsingular Hermitian matrix P such that AB is similar to . Since is Hermitian, it is similar to a real diagonal matrix. Hence, AB is similar to a real diagonal matrix.Exercise 10Let A be an Hermitian matrix of order n. Show that there exists a real number such that is positive definite. Proof 1: The matrix is Hermitian for real values of t. If the eigenvalues of A are , then the eigenvalues of are . Let Then the eigenvalues of are all positive. And hence, is positive definite.Proof 2: The matrix is Hermitian for real values of t. Let be the leading principle minor of A of order r. terms involving lower powers in t. Hence, is positive for sufficiently large t. Thus, if t is sufficiently large, all leading principal minors of will be positive. That is, there exists a real number such that is positive for and for each r. Thus is positive definite for .Exercise 11 Let be an Hermitian positive definite matrix. Show that Proof We first prove that if A is Hermitian positive definite and B is Hermitian semi-positive definite, then . Since A is positive definite, there exists a nonsingular hermitian matrix P such that is positive semi- definite. Its eigenvalues are all greater than or equal to 1. Thus is positive definite, and is positive semi-definite, and Hence, This finishes the proof. Exercise 12Let A be a positive definite Hermitian matrix of order n. Show that the element in A with the largest norm must be in the main diagonal.Proof Let . Suppose that is of the largest norm, where . Consider the principal minor . It must be positive definite since A is positive definite. (Recall that an Hermitian matrix is positive definite iff all its principal mino