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a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$.

I have $ab^TCab^TC$.

I try to manipulate this multiplication into:

$b^TCaab^TC$.

I need help.

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1 Answer 1

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All you need here is to realize that $b^{T}Ca$ is a scalar, it is the dot product of the vector $b$ with the vector $Ca$. As it is a scalar, it can be factored out of the expression. This gives the form you want. In particular, $$ab^{T}Cab^{T}C = a(b^{T}Ca)b^{T}C = (b^{T}Ca)ab^{T}C.$$

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