Complex Matrices In this project we assume that you have either had some experience with m...

Complex Matrices In this project we assume that you have either had some experience with matrices or are willing to learn something about them.

Certain complex matrices, that is, matrices whose entries are complex numbers, are important in applied mathematics. An n × n complex matrix A is said to be:

Here the symbol Ā means the conjugate of the matrix A, which is the matrix obtained by taking the conjugate of each entry of A. ĀT is then the transpose of Ā, which is the matrix obtained by interchanging the rows with the columns. The negative − A is the matrix formed by negating all the entries of A; the matrix A −1 is the multiplicative inverse of A.

(a) Which of the following matrices are Hermitian, skew-Hermitian, or unitary?

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(b) What can be said about the entries on the main diagonal of a Hermitian matrix? Prove your assertion.

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(c) What can be said about the entries on the main diagonal of a skew-Hermitian matrix? Prove your assertion.

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(d) Prove that the eigenvalues of a Hermitian matrix are real.

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(e) Prove that the eigenvalues of a skew-Hermitian matrix are either pure imaginary or zero.

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(f) Prove that the eigenvalues of unitary matrix are unimodular; that is, |λ| = 1. Describe where these eigenvalues are located in the complex plane.

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(g) Prove that the modulus of a unitary matrix is one, that is, |detA| = 1.

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(h) Do some additional reading and find an application of each of these types of matrices.

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(i) What are the real analogues of these three matrices?