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LAHW#16 •Due ∞ 8.1 Hermitian Matrices and the Spectral Theorem • 3. – Find the eigenvalues of this matrix: 2 3i 1 i 2 i 1 2i 8.1 Hermitian Matrices and the Spectral Theorem • 9. – Explain why the diagonal of a Hermitian matrix is real. 8.1 Hermitian Matrices and the Spectral Theorem • 27. – Let L be a self-adjoint operator on a finitedimensional inner-product space (the scalar field being R). Let A be the matrix for L with respect to some basis. Is A necessarily symmetric? A theorem or example is needed. 8.1 Hermitian Matrices and the Spectral Theorem • 33. – Let {v1, v2, …, vn} ben points in an inner-product space. Define T ( x) ri x, vi vi i 1 where r1, r2, …, rn, are real numbers. Establish that T us self-adjoint. 8.1 Hermitian Matrices and the Spectral Theorem • 36. – Establish that if the matrix A is Hermitian, then Ax, x is real valued. the quadratic form x Assume that x ranges over the space Cn. 8.1 Hermitian Matrices and the Spectral Theorem • 39. – Use the Cayley-Hamilton Theorem to get a formula for A-1 when A is invertible. Be sure to use some where in your work the hypothesis that A is invertible! You can assume that the characteristic equation of A is known. 8.1 Hermitian Matrices and the Spectral Theorem • 41. – Explain why, for any n×n matrix A, An is in the span of { I, A, A2, …, An-1}. 8.1 Hermitian Matrices and the Spectral Theorem • 51. – For z z1, z2 ,..., zn C , we use the notation z z1 , z2 ,..., zn . Is the mapping z z linear?