Let x,w∈Rn How to show that: (wtx)x=(xxt)w (t denotes transpose) Attempt at proof: α=(wtx)x αt=xt(wtx)t=xt(xtw) (αt)t=(xt)t(xtw) (treating xt w as a scalar here) α=x(xtw)=(xxt)w (associativity)

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Lynne Trussell · Accepted Answer

Your proof looks correct to me, you are using that xTw is a scalar, and also that the matrix multiplication is associative. More directly you can argue that (wTx)x=x(wTx)=x(xTw)=(xxT)w where the first two steps use that wTx is a scalar, and the last step uses that the matrix multiplication is associative.

Annie Gonzalez · Accepted Answer

I guess that v′ denotes the transpose.This uses a trick, besides the transposition: w′x is a scalar to be multiplied by an n×1 matrix. You get the same result if you instead consider the product x(w′x) where now w′x is considered as an 1×1 matrix. But w′x=x′w, so you can write(w′x)x=(x′w)x=x⏟n×1(x′w)⏟1×1=(xx′)⏟n×nw⏟n×1

Let PSKx,w \in \mathbb{R}^nZSKNSKHow to show that: PSK(w^t x)x=(xx^t)wZSK

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