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Exercise 1.10.8

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$S = u u^{T}$ , then $R (x) = \frac{x^{T} Sx}{x^{T} Mx} = \frac{x^{T} u u^{T} x}{x^{T} Mx} = \frac{{(u^{T} x)}^{T} (u^{T} x)}{x^{T} Mx} = = \frac{{(u^{T} x)}^{2}}{x^{T} Mx}$ since $u^{T} x$ is a scalar. It’s clear that when $M$ is positive definite, $R (x)$ achieves minimum value of 0 when $x$ is perpendicular to $u$ .

niuers

2020-03-20 00:00

Exercise 1.10.8

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