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Exercise 1.8.22

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$A = U_{r} Σ_{r} V_{r}^{T}$ , for a given $v_{j}$ , we have

$A v_{j} = U_{r} Σ_{r} V_{r}^{T} v_{j} = U_{r} Σ_{r} [\begin{matrix} 0 \\ 0 \\ \dots \\ 1 \\ \dots \\ 0 \end{matrix}] = U_{r} [\begin{matrix} 0 & \dots & 0 & \dots & 0 \\ 0 & \dots & σ_{j} & \dots & 0 \\ 0 & \dots & 0 & \dots & 0 \end{matrix}] = σ_{j} u_{j}$ , which proves that $A$ satisfies equation (1).

niuers

2020-03-20 00:00

Exercise 1.8.22

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