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Problem 6 Let U be an upper triangular n X n matrix Prove thatK(U) max |Uii|/ min |Uiil 1<i<n 1Ki<nUse this inequality to estimate the condition number ...

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Problem 6 Let U be an upper triangular n X n matrix Prove thatK(U) max |Uii|/ min |Uiil 1<i<n 1Ki<nUse this inequality to estimate the condition number of~2 1 0 010 05 6 7 5 10A =Compare with the true condition number of that matrix

Problem 6 Let U be an upper triangular n X n matrix Prove that K(U) max |Uii|/ min |Uiil 1<i<n 1Ki<n Use this inequality to estimate the condition number of ~2 1 0 0 10 0 5 6 7 5 10 A = Compare with the true condition number of that matrix



Answers

Let $A$ be an $n \times n$ symmetric matrix, let $M$ and $m$ denote the maximum and minimum values of the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ , where $\mathbf{x}^{T} \mathbf{x}=1,$ and denote corresponding unit eigenvectors by $\mathbf{u}_{1}$ and $\mathbf{u}_{n} .$ The following calculations show that given any number $t$ between $M$ and $m,$ there is a unit vector $\mathbf{X}$ such that $t=\mathbf{x}^{T} A \mathbf{x} .$ Verify that $t=(1-\alpha) m+\alpha M$ for some number
$\alpha$ between 0 and $1 .$ Then let $\mathbf{x}=\sqrt{1-\alpha} \mathbf{u}_{n}+\sqrt{\alpha} \mathbf{u}_{1},$ and show that $\mathbf{x}^{T} \mathbf{x}=1$ and $\mathbf{x}^{T} A \mathbf{x}=t$


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