Page 314 - Advanced engineering mathematics

294 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

In each of Problems 22 through 28, determine a matrix A 28. −2x 1 x 2 + 2x 2
2
t
so that the quadratic form is X AX, and find the standard
29. Suppose A is hermitian. Show that
form of the quadratic form.
(AA ) = AA.
t
22. −5x + 4x 1 x 2 + 3x 2 2
2
1
2
23. 4x − 12x 1 x 2 + x 2 2 30. Prove that the main diagonal elements of a hermitian
1
matrix are real.
2
24. −3x + 4x 1 x 2 + 7x 2 2
1
31. Prove that each main diagonal element of a skew-
2
25. 4x − 4x 1 x 2 + x 2 2 hermitian matrix is zero or pure imaginary.
1
26. −6x 1 x 2 + 4x 2
2 32. Prove that the product of two unitary matrices is
2
27. 5x + 4x 1 x 2 + 2x 2 unitary.
1 2

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October 14, 2010 14:49 THM/NEIL Page-294 27410_09_ch09_p267-294