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I have a 6x6 matrix that equals the original matrix when multiplied by its transpose. What does this say about this matrix? What unique conditions does this matrix satisfy, since this property doesn't seem to hold in general?

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I don't follow. This means that the matrix is the identity. Did you mean to say something else? – Qiaochu Yuan Oct 14 '12 at 2:08
Sorry, I made a mistake. I meant matrix times its transpose, not matrix times its inverse. – metrix Oct 14 '12 at 2:09
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In the case of a symmetric matrix, this property is called idempotence. – Max Oct 14 '12 at 2:11
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The math working out is a reason. Is there anything in particular you're after? – EuYu Oct 14 '12 at 2:13
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This certainly doesn't seem like a lucky coincidence to me. Perhaps you should include more details of the context of the problem. – EuYu Oct 14 '12 at 2:17
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If I understand the comments correctly, the matrices you're interested in have the following two properties:

  • They are symmetric: $M^T = M$.
  • They are idempotent: $M^2 = M$.

Let me further assume that your matrices are real. Then these matrices are precisely the orthogonal projections onto some subspace (namely their image).

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