Could you please help with question 306 and 307? Thanks

Exercise 293 Exercise 294 Normal matrix Exercise 295 Symmetry Exercise 296 Exercise 297 Thcorcm 2 73 5.7 Localization of eigenvalues One of the most important questions is how does f (A) change when we perturb A. We already considered this question when f (1:) = 221 in Section 3.7. An obvious idea is to use the Jordan decomposition to help make this estimate. For example ||f(A)| = |f(VJV1)|| S |V|||V_1||||f(J)l- But this upper bound can be wildly inaccurate if H.[V) = ||V||||V_1| is very large. However, better general-purpose estimates are hard to come by. So one approach is to look for special classes of matrices for which K.(V) is small in a suitable norm. Let n2(V) = |V||2|V_1||2 denote the 2-norm condition number of the matrix V. Show that s2 (V) 2 1. Hint: Use the SVD. Show that if V is a unitary matrix then H.2(V) = 1. A matrix A is said to be normal if AAH = AHA. 100 Show that unitary and orthogonal matrices are normal. A matrix A is said to be 101 a symmetric if AT = A o skew-symmetric if AT = A o Hermitian if AH = A o skew-Hermitian if AH = A Show that Hermitian and skewHermitian matrices are normal. Show that every matrix can be written uniquely as the sum of a Hermitian and a skew-Hermitian matrix. Hint: A AH A AH A = + + _. 2 2 Let A be normal. Then there exists a unitary matrix Q and a diagonal matrix A such that A = QAQH. In other words for normal matrices the Schur decomposition is also the Jordan decomposition with each Jordan block being of size one. Furthermore there is a full set of orthonormal eigenvectors. Lemma 8 Exercise 298 Exercise 299 Exercise 300 Exercise 301 Exercise 302 Exercise 303 Exercise 304 Exercise 305 Exercise 306 Exercise 307 Exercise 308 74 Proof. The proof follows from the following fact. If R is an upper triangular normal matrix then it is diagonal. Prove the lemma. Hint: Write R as a 2 x 2 block matrix and solve the four resulting equations. Prove the theorem. CI It follows that for normal matrices ||f(A)|| = ||f(A)||, where the diagonal entries of A are the eigenvalues of A. Therefore it becomes essential to locate, at least approximately, the eigenvalues of A in the complex plane. Let A be normal with Schur decomposition A = QAQH . Consider the expression (QHAQ)H and use it to prove the next three exercises. Show that the eigenvalues of a unitary matrix must lie on the unit circle. Show that the eigenvalues of a Hermitian matrix must be real. Show that the eigenvalues of a skew-Hermitian matrix must be purely imaginary. Show that the eigenvectors of a normal matrix corresponding to distinct eigenvalues must be mutually orthogonal. Hint: Use the Schur decomposition. Write down a family of normal matrices that is neither unitary nor Hermitian nor skew-Hermitian. Hint: Use the Schur decomposition. Show that es'k'iw'Hmm'tiarl = unitary. Show that eiHem'j'L'i'11 = unitary. Let A be a square real matrix. Suppose /\\ is an eigenvalue of A with a non-zero imaginary part. a Show that the corresponding eigenvector v, must have real and imaginary parts that are linearly independent when considered as real vectors.. a Show that :\\ must also be an eigenvalue of A. a Show that an eigenvector for is can be constructed from v. Show that a real orthogonal matrix with an odd number of rows and columns must have either 1 or 1 as one of its eigenvalues