Show that a matrix is:

i) Hermitian if and only if it can be diagonalized using an orthonormal basis of eigenvectors and the eigenvalues are real;

ii) skew Hermitian if and only if it can be diagonalized using an orthonormal basis of eigenvectors and the eigenvalues are imaginary;

iii) positive definite if and only if it can be diagonalized using an orthonormal basis of eigenvectors and the eigenvalues are real and positive;

iv) unitary if and only if it can be diagonalized using an orthonormal basis of eigenvectors and the eigenvalues are on the unit circle;

v) normal if and only if it can be diagonalized using an orthonormal basis of eigenvectors.

See questions 5.4.36 - 5.4.44 for definitions and ideas on how to solve these problems.

Exercise 5.4.36 Recall that a Hermitian matrix is called positive definite if for all nonzero x e Cn, x* Ax > 0. Prove that if A is positive definite and B is unitarily similar to A, then B is also positive definite.

Exercise 5.4.37 Use Rayleigh quotients to prove that the eigenvalues of a positive definite matrix are positive. The next exercise indicates a second way to prove this.

Exercise 5.4.38 Let A £ Cnx" be a Hermitian matrix. Use Theorem 5.4.12 and the result of Exercise 5.4.36 to prove that A is positive definite if and only if all of its eigenvalues are positive.

Exercise 5.4.39 A Hermitian matrix A £ Cnxn is positive semidefinite if x*Ax > 0 for all x £ C". Formulate and prove results analogous to those of Exercises 5.4.36 to 5.4.38 for positive semidefinite matrices.

Exercise 5.4.40 A matrix A £ (Cnxn is skew Hermitian if A* = -A. (a) Prove that if A is skew Hermitian and B is unitarily similar to A, then B is also skew Hermitian. (b) What special form does Schur's theorem (5.4.11) take when A is skew Hermi¬ tian? (c) Prove that the eigenvalues of a skew Hermitian matrix are purely imaginary; that is, they satisfy A = — A. Give two proofs, one based on Schur's theorem and one based on the Rayleigh quotient.

Exercise 5.4.41 (a) Prove that if A is unitary and B is unitarily similar to A, then B is also unitary. (b) Prove that a matrix T £ £nxn that is both upper triangular and unitary must be a diagonal matrix. (You will prove a more general result in Exercise 5.4.42.) (c) What special form does Schur's theorem (5.4.11) take when A is unitary? (d) Prove that the eigenvalues of a unitary matrix satisfy A = A-1. Equivalently;

AA = 1 or | A |2 = 1; that is, the eigenvalues lie on the unit circle in the complex plane. Give two proofs.

Exercise 5.4.42 A matrix A £ cnxn is normal if AA* = A*A. (a) Prove that all Hermitian, skew-Hermitian, and unitary matrices are normal. (b) Prove that if A is normal and B is unitarily similar to A, then B is also normal. (c) Prove that a matrix T £ £nxn that is both upper triangular and normal must be a diagonal matrix. (Hint: Use induction on n. Write T in the partitioned form

T=[t_{11}, s^{*}; 0, T(hat)]

then write down the equation TT* = T*T in partitioned form and deduce that s = 0 and T is also (triangular and) normal.) (d) Prove that every diagonal matrix is normal. (e) Prove Theorem 5.4.16: A is normal if and only if A is unitarily similar to a diagonal matrix.

Exercise 5.4.43 Let D e Cnx" be a diagonal matrix. Show that... (a) D is Hermitian if and only if its eigenvalues are real. (b) D is positive semidefinite if and only if its eigenvalues are nonnegative. (c) D is positive definite if and only if its eigenvalues are positive. (d) D is skew Hermitian if and only if its eigenvalues lie on the imaginary axis of the complex plane. (e) D is unitary if and only if its eigenvalues lie on the unit circle of the complex plane.

Exercise 5.4.44 Let A G €nxn be a normal matrix. Show that... (a) A is Hermitian if and only if its eigenvalues lie on the real axis. (b) A is positive semidefinite if and only if its eigenvalues are nonnegative. (c) A is positive definite if and only if its eigenvalues are positive. (d) A is skew Hermitian if and only if its eigenvalues lie on the imaginary axis. (e) A is unitary if and only if its eigenvalues lie on the unit circle.

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