Q1 Hadamard's inequality (i) Let A = (a lan) E Rnxn. Let A = QR =...
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Q1 Hadamard's inequality (i) Let A = (a₁ lan) E Rnxn. Let A = QR = Q(r₁|rn) be a QR factorization of A. - First, show that det(A)| ≤II ri||2. Then, using the previous part, show that det(A)| ≤ II1 |ai||2. This inequality is known as "Hadamard's inequality". - (ii) Let A = (a,j) € Rnxn with aj € {-1, 1} Vi, je {1,...,n}. Using (i), show that det(A)| ≤ n. (iii) Show that if A € Rx" is a Hadamard matrix, then [det(A) = n*. Q1 Hadamard's inequality (i) Let A = (a₁ lan) E Rnxn. Let A = QR = Q(r₁|rn) be a QR factorization of A. - First, show that det(A)| ≤II ri||2. Then, using the previous part, show that det(A)| ≤ II1 |ai||2. This inequality is known as "Hadamard's inequality". - (ii) Let A = (a,j) € Rnxn with aj € {-1, 1} Vi, je {1,...,n}. Using (i), show that det(A)| ≤ n. (iii) Show that if A € Rx" is a Hadamard matrix, then [det(A) = n*.
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i Let A QR be the QR factorization of A where Q is an orthogonal matrix and R is an upper triangular matrix Since Q is orthogonal its determinant is either 1 or 1 Therefore we have detA detQR detQ det... View the full answer
Related Book For
Probability and Statistics
ISBN: 978-0321500465
4th edition
Authors: Morris H. DeGroot, Mark J. Schervish
Posted Date:
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