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There are a number of places this matrix transform making its appearance:

  1. Every positive semi-definite matrix $B$ can have a decomposition $B=AA^*$

  2. If the matrix $A$ is a lower triangular matrix then $B=AA^*$ is also called as Cholesky decomposition of $B$

  3. Given a system of linear equations: $Ax=b$, $(A^*A)x = (A^*b)$ represents "normal equations"

  4. If X is a random vector, $\mathbb{E}[XX^*]$ is called its covariance matrix

  5. Poalr decomposition problem : A = UP (Unitary and Positive semi-definite) is solved by $P=\sqrt{A^*A}$ (Courtesy: Jonas Meyer)

... and so on.

I suspect there is a name for the tranformation $A^*A$ or $AA^*$!

Is there a name really? What would you suggest if there is n't?

A matrix of normal equations ?? (Doesn't look nice!)

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  • $\begingroup$ By analogy with complex numbers, it can be thought of as the square of the absolute value of $A$. One problem with that is that $A^*A$ could equally well be called the same. The analogy works well if and only if $A$ is normal. (Another example where it is used is in the polar decomposition.) $\endgroup$ – Jonas Meyer Mar 11 '15 at 20:15
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    $\begingroup$ The Grammian or Gram Matrix looks an awful lot like what you're asking for. $\endgroup$ – Randy E Mar 11 '15 at 20:45
  • $\begingroup$ @RandyE is right. But that requires that it be thought of in terms of the row or column vectors. Something like "the Grammian of the columns of $A$". I guess what you call it might depend on which of the applications you're in. $\endgroup$ – Jonas Meyer Mar 11 '15 at 21:15
  • $\begingroup$ @RandyE Thanks a lot!! Exactly like what I need. I will accept this if you can post it as an answer. In fact, combined with Jonas Meyer's suggestion, we have a way to name both $A^*A$ and $AA^*$ $\endgroup$ – Loves Probability Mar 11 '15 at 21:51
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The Gramian or Gram Matrix looks like what you're asking for. As JonasMeyer points out, you have to take into consideration whether you want the Gram matrix of the row or the column vectors.

(Subjective addendum: To simplify terminology, instead of saying "the Gramian of the columns of $A$" and "the Gramian of the rows of $A$," you could call $A^TA$ and $AA^T$ something like the column-Gramian and the row-Gramian, respectively. You could take the simplification one step further by calling one of these the Gramian, and only requiring the prefix for the other one (similar to what we do with eigenvectors: there are right and left eigenvectors, but "eigenvector" on its own usually means right eigenvector.))

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