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We define A to be a matrix in $R^{m*n}$

  1. Does $A^TA$ have any particular structure?
  2. When is $A^TA$ invertible?
Joe
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  • I know that the transpose of a matrix times itself is square and symmetric. Any help for the second? – Joe Jul 22 '14 at 04:47

1 Answers1

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Some of the properties of the matrix you are talking about are

  • In my area of research (signal processing), this is referred to as the gram matrix. I believe, this is the standard name as well.
  • It is a positive (semi) definite (PD) matrix. If you know about PD matrices, proving this is straight forward. The trick is this relation $x^Tx=||x||_2^2\geq 0$. Thus all results applicable to PD matrices are also applicable to the above matrix. (For eg: all eigenvalues are non-negative, cholesky decomposition and so on).
  • Then, if $det(A^TA)\neq 0$, then columns of $A$ are linearly independent. Try to think about the converse.
dineshdileep
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  • Great help! thank you. – Joe Jul 22 '14 at 05:20
  • I believe the gram matrix is $A A^T$, not $A^T A$. These are equivalent if (and only if) the matrix is normal – Pro Q Sep 13 '22 at 19:21
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    @ProQ, well, it will depend on how you define the data vectors, whether as column or row of $A$. The prevalent definition seems to be what I did.

    https://en.wikipedia.org/wiki/Gram_matrix

     – dineshdileep Sep 13 '22 at 20:43