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I have a matrix-vector inner product multiplication $G = X D x$ where $D$ is a diagonal matrix. Now let's say I already know $E = Xx$. Is there a method that I can use to change $E$ into $G$ using $D$ without having to calculate $G$ in full?

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Merely knowing the vector $E$ and the matrix $D$ doesn't seem to be enough to determine $G$, since $E$ is some vector in the column space of $X$, and $G$ can be more or less any other vector in that column space. I don't see how you can tell what $G$ is without using what the individual columns in $X$ are and how they were combined to obtain $E$.

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I love cheap answers because they help in asking good questions. So you have $G=XDX^{-1}E$ How do you intend to use $G$ that this does not satisfy? I don't understand where diagonal comes into this.

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To clarify: I know $E$, but I do not want to use $X$ and $x$ in further calculations if possible. Diagonal may not be relevant, just mentioned it in case it could simplify the problem. – Projectile Fish Feb 3 '11 at 6:10

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