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Let n be a positive integer, A an n x n matrix, and x € R". View x as a column vector, i.e., an n x 1 matrix. (a) Using rules of transposition, show that (xT Ax) = xTATx. Explain carefully how the rules of transposition are used. Failure to do so may result in lost marks. (b) By considering the sizes of x¹, A, and x, explain briefly why xAx is a 1 x 1 matrix. Since every 1 x 1 matrix is symmetric, it follows that (xT Ax) = x Ax. (c) By combining parts (a) and (b), show that x¹ Ax = xTBx, where B is the symmetric matrix (A + AT). This fact is useful in the theory of quadratic forms.

Let n be a positive integer, A an n x n matrix, and x € R". View x as a column vector, i.e., an n x 1 matrix. (a) Using rules of transposition, show that (xT Ax) = xTATx. Explain carefully how the rules of transposition are used. Failure to do so may result in lost marks. (b) By considering the sizes of x¹, A, and x, explain briefly why xAx is a 1 x 1 matrix. Since every 1 x 1 matrix is symmetric, it follows that (xT Ax) = x Ax. (c) By combining parts (a) and (b), show that x¹ Ax = xTBx, where B is the symmetric matrix (A + AT). This fact is useful in the theory of quadratic forms.

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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Let n be a positive integer, A an n x n matrix, and x € R". View x as a column vector, i.e., an n x 1 matrix. (a) Using rules of transposition, show that (xAx) = xTATx. Explain carefully how the rules of transposition are used. Failure to do so may result in lost marks. (b) By considering the sizes of x¹, A, and x, explain briefly why x Ax is a 1 x 1 matrix. Since every 1 x 1 matrix is symmetric, it follows that (xT Ax) = x Ax. (c) By combining parts (a) and (b), show that x¹ Ax = xTBx, where B is the symmetric matrix (A + AT). This fact is useful in the theory of quadratic forms.
Transcribed Image Text:Let n be a positive integer, A an n x n matrix, and x € R". View x as a column vector, i.e., an n x 1 matrix. (a) Using rules of transposition, show that (xAx) = xTATx. Explain carefully how the rules of transposition are used. Failure to do so may result in lost marks. (b) By considering the sizes of x¹, A, and x, explain briefly why x Ax is a 1 x 1 matrix. Since every 1 x 1 matrix is symmetric, it follows that (xT Ax) = x Ax. (c) By combining parts (a) and (b), show that x¹ Ax = xTBx, where B is the symmetric matrix (A + AT). This fact is useful in the theory of quadratic forms.
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