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1. Let V be an n-dimensional inner product space and let {u₁, 2,. for V. If v=au, and w= bu, where a, b, EF, then (v, w) = i=1 Un} be an orthonormal basis 72 a,b,. i=1

1. Let V be an n-dimensional inner product space and let {u₁, 2,. for V. If v=au, and w= bu, where a, b, EF, then (v, w) = i=1 Un} be an orthonormal basis 72 a,b,. i=1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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VI. Prove the following statements. 1. Let V be an n-dimensional inner product space and let {u₁, 2,..., un} be an orthonormal basis n for V. If v=au, and w =) bui, where a, b, EF, then (v, w) = abi. i=1 i=1 i=1 2. If a matrix A is diagonalizable, show that A² = PD²P-¹ for some nonsingular matrix P and diagonal matrix D. Find an expression for A" where n is a positive integer. 3. If A € M₂, (C) is skew-Hermitian then the eigenvalues of A are either zero or pure imaginary.
Transcribed Image Text:VI. Prove the following statements. 1. Let V be an n-dimensional inner product space and let {u₁, 2,..., un} be an orthonormal basis n for V. If v=au, and w =) bui, where a, b, EF, then (v, w) = abi. i=1 i=1 i=1 2. If a matrix A is diagonalizable, show that A² = PD²P-¹ for some nonsingular matrix P and diagonal matrix D. Find an expression for A" where n is a positive integer. 3. If A € M₂, (C) is skew-Hermitian then the eigenvalues of A are either zero or pure imaginary.
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