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Linear Algebra and Learning from Data

Exercise 1.10.6

Answers

1.
Because the transpose of the congruence $Z^{T} SZ$ is itself.
1.
For a given $x \neq 0$ , we have $x^{T} Z^{T} SZx = {(Zx)}^{T} S (Zx)$ , since $S$ is positive definite so let $y = Zx$ , we have $x^{T} Z^{T} SZx = y^{T} Sy$ . If $Z$ is square and invertible, it means its rank equals to the number of columns. So the nullspace is empty. So when $x \neq 0$ , we have $Zx \neq 0$ . We now see that the congruence matrix is also positive definite.

niuers

2020-03-20 00:00

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