A Hessenberg matrix H, is n x n matrix that has 2 on the main diagonal, 1everywhere below the main diagonal and on the diagonal above the mainone and zeros everywhere else. For example,21H212[2 101H3 = | 1 2121 2 11 1 2 1H41 1 1 21a)Compute det(H2) and det(H3).1b)Compute det(H4) by doing the following steps: subtract column 2 fromcolumn 1 and use cofactor expansion along the resulting column 1 to showthatdet(H4) = det(H3) + det(H2).1c) Follow the steps in (1b) to show that the same result holds for arbitraryn > 3det(H„) = det(H,-1)+ det(Hn-2).

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A Hessenberg matrix H„ is n × n matrix that has 2 on the main diagonal, 1 everywhere below the main diagonal and on the diagonal above the main one and zeros everywhere else. For example, 2 H2 [1 2 2 1 0 H3 = | 1 2 1 1 1 2. 2 100 1 2 1 0 1 1 2 1 1 1 1 2. H4 = ||

A Hessenberg matrix H„ is n × n matrix that has 2 on the main diagonal, 1 everywhere below the main diagonal and on the diagonal above the main one and zeros everywhere else. For example, 2 H2 [1 2 2 1 0 H3 = | 1 2 1 1 1 2. 2 100 1 2 1 0 1 1 2 1 1 1 1 2. H4 = ||

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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Question

A Hessenberg matrix H, is n x n matrix that has 2 on the main diagonal, 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 1
01
H3 = | 1 2
1
2
1 2 1
1 1 2 1
H4
1 1 1 2
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(H„) = det(H,-1)+ det(Hn-2).

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