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hw01_solution

# hw01_solution - EEL 6935 Electronic Navigation Systems...

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EEL 6935 Electronic Navigation Systems Homework Solution #1 Problem 1 Show that for any orthonormal matrix A , | A | = ± 1 HINT: Use A T A = I . Solution: For any orthonormal matrix A : A T A = I | A T || A | = | I | But for any matrix, | A T | = | A | , so | A | 2 = 1 | A | = ± 1 Problem 2 Show that for any orthonormal matrix A , k x k = k A x k HINT: Express x as a weighted sum of columns of A , then calculate x T x . Solution: The product A x can be written as A x = a 1 a 2 · · · a N x 1 x 2 . . . x N = a 1 x 1 + a 2 x 2 + · · · + a N x N , 1

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where { a 1 , a 2 , · · · , a N } are the orthonormal set of columns of A . k A x k 2 = ( A x ) T A x = ( a 1 x 1 + a 2 x 2 + · · · + a N x N ) T ( a 1 x 1 + a 2 x 2 + · · · + a N x N ) = a T 1 a 1 x 2 1 + a T 1 a 2 x 1 x 2 + · · · + a T 1 a N x 1 x N + a T 2 a 1 x 2 x 1 + a T 2 a 2 x 2 2 + · · · + a T 2 a N x 2 x N · · · + a T N a 1 x N x 1 + a T N a N x 2 2 + · · · + a T N a N x 2 N = x 2 1 + x 2 2 + · · · + x 2 N = k x k 2 Problem 3 Suppose that the random vector X has covariance matrix C = σ 2 1 0 0 σ 2 2 .
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hw01_solution - EEL 6935 Electronic Navigation Systems...

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