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?
Matrix times its transpose equals original matrix
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1I don't follow. This means that the matrix is the identity. Did you mean to say something else? – 2012-10-14
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0Sorry, I made a mistake. I meant matrix times its transpose, not matrix times its inverse. – 2012-10-14
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1In the case of a symmetric matrix, this property is called [idempotence](http://en.wikipedia.org/wiki/Idempotent_matrix). – 2012-10-14
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0Yes, the matrix I'm talking about is symmetric. Is this simply the way the math works out, or are there any reasons and other properties for this happening? – 2012-10-14
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1The math working out **is** a reason. Is there anything in particular you're after? – 2012-10-14
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0I'm looking at these in the context of regression coefficients, and I wonder if this means anything special, or is it simply a lucky convenience – 2012-10-14
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1This certainly doesn't seem like a lucky coincidence to me. Perhaps you should include more details of the context of the problem. – 2012-10-14
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0It's not actually a problem, but I was looking at some econometrics notes and noticed that the "residual maker" matrix and OLS projection matrix are both symmetric and idempotent. I understand that the math works out, but not what it means graphically or intuitively in terms of regressions. – 2012-10-14
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0Thank you. I see it now. Maybe I didn't have enough reputation before - it just increased. – 2012-10-14
1 Answers
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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).