Finding a transformation that squares the components of a vector

Question

I am looking for a transformation (of any type, as long as it can be written mathematically and not semantically) which takes a vector of infinite dimensions to another vector that has components which are squares are the first. That is to say,

$T(x_1,x_2,x_3...) = (x_1^2,x_2^2,x_3^2...)$

for such a transformation T.

My very first knee-jerk thought was that I could take the dot product of my vector with itself, or else multiply the transpose of my vector with itself, because these both have the property of multiplying each component of the vector itself, but then both do the annoying thing and then sum all of these together.

My next thought was to construct an infinite diagonal matrix where each entry was equal to the corresponding component in the vector, but I couldn't find an easy way to construct this matrix algebraically for the general vector.

Any speculation about whether this question is possible or even makes sense is more than welcome.

Answer 1

Use

$$\boldsymbol{y}= {\rm diag}( \boldsymbol{x} \boldsymbol{x}^\intercal )$$

Which takes the diagonal elements matrix formed from the outer product of $\boldsymbol{x}$ with itself.