An n x n complex matrix A is called skew-hermitian if and only if: A =...
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An n x n complex matrix A is called skew-hermitian if and only if: A¹ = -A. A real, skew-hermitian matrix is called anti-symmetric or skew-symmetric. a) Let A be a skew-hermitian matrix and suppose A E C is an eigen- value of A. Prove that λ = ib, with b E R. (Hint: Mimic the proof done in class to show that the eigenvalues of a Hermitian matrix are all real.) b) Let A be a real, skew-symmetric matrix, prove that if X ER is an eigenvalue, then λ = 0. c) Prove that if n is odd, then 0 is always an eigenvalue of A. d) Find all eigenvalues and eigenvectors for the skew-symmetric matrices: = (-28) A₁ = = A₂ 42 = ( 3 0 -3 0-4 4 0 An n x n complex matrix A is called skew-hermitian if and only if: A¹ = -A. A real, skew-hermitian matrix is called anti-symmetric or skew-symmetric. a) Let A be a skew-hermitian matrix and suppose A E C is an eigen- value of A. Prove that λ = ib, with b E R. (Hint: Mimic the proof done in class to show that the eigenvalues of a Hermitian matrix are all real.) b) Let A be a real, skew-symmetric matrix, prove that if X ER is an eigenvalue, then λ = 0. c) Prove that if n is odd, then 0 is always an eigenvalue of A. d) Find all eigenvalues and eigenvectors for the skew-symmetric matrices: = (-28) A₁ = = A₂ 42 = ( 3 0 -3 0-4 4 0
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Related Book For
Introduction to Algorithms
ISBN: 978-0262033848
3rd edition
Authors: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest
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