A Hessenberg matrix H„ is n x n matrix that has 2 on the main diaonal, 1 everywhere below the main diagonal and on the diagonal above the main one and zeros everywhere else. For example, [2 1 H2 = |1 2 2 107 H3 = 1 2 1 [1 1 2 2 10 0 1 2 1 0 1 1 2 1 [i 1 1 2. H4 (1a, ) Compute det(H2) and det(H3) (1b, ) Compute det(H4) by doing the following steps: subtract column 2 from column 1 and use cofactor expansion along the resulting column 1 to show that det(H4) = det(H3) + det(H2). (1c, ) Follow the steps in (1b) to show that the same result holds for arbitrary n > 3 det(Hn) = det(H„-1) + det(Hn-2).

A Hessenberg matrix H„ is n x n matrix that has 2 on the main diaonal, 1 everywhere below the main diagonal and on the diagonal above the main one and zeros everywhere else. For example, [2 1 H2 = |1 2 2 107 H3 = 1 2 1 [1 1 2 2 10 0 1 2 1 0 1 1 2 1 [i 1 1 2. H4 (1a, ) Compute det(H2) and det(H3) (1b, ) Compute det(H4) by doing the following steps: subtract column 2 from column 1 and use cofactor expansion along the resulting column 1 to show that det(H4) = det(H3) + det(H2). (1c, ) Follow the steps in (1b) to show that the same result holds for arbitrary n > 3 det(Hn) = det(H„-1) + det(Hn-2).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 23EQ

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