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There is a sentense that says that every symmetric matrix is congruent to a diagonal matrix.

I've been trying to find the congruent matrix and the transition matrix for the following: $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

The method I learned to get to a solution is to do row operations on both the matrix and the identity matrix, and the according column operations on the original matrix only until I get to a diagonal matrix from the original and the transition matrix from the identity matrix.

This process seems to loop infinitly in with this matrix when I switch rows 2 and 3.

So I need to know how to find the congruent diagonal matrix and the transition matrix for the given matrix and a method which will be fail-proof.

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  • $\begingroup$ Hint: Every diagonlizable matrix has a basis of eigenvectors $\endgroup$ – Oria Gruber Apr 5 '14 at 20:11
  • $\begingroup$ @OriaGruber I'm searching for a congruency relation, not similarity. $\endgroup$ – NightRa Apr 5 '14 at 20:15
  • $\begingroup$ I realize that. but - because you have an orthonormal basis of eigenvectors, $P^{-1}=P^T$ so if you find matrix $P$ of eigenvectors such that $P^{-1}AP=D$, because $P$ is an ORTHOGONAL matrix (because $A$ is symmetric) you get $P^TAP=D$ $\endgroup$ – Oria Gruber Apr 5 '14 at 20:16
  • $\begingroup$ @OriaGruber I think that the similarity requirement is stronger than congruency, and it could be done more easily. I think I am just missing a detail with the method we were taught. $\endgroup$ – NightRa Apr 5 '14 at 20:18
  • $\begingroup$ Realize that in the case of symmetric matrices, similarity and congruency are the same thing. $\endgroup$ – Oria Gruber Apr 5 '14 at 20:22
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Rather than switching the columns, add the third row to the second, and work from there.

I am not familiar with the algorithm you describe, and so I can't tell you why switch the rows would cause it to fail. However, you should find that $$ P = \pmatrix{1&0&0&0\\ 0&1&1&0\\ 0&1&-1&0\\ 0&0&0&1} $$ will work for your purposes (incidentally, note that $P = P^T$).

In terms of row-column operations, here's how it would go: $$ \pmatrix{1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1} \quad \text{...II = II + III}\\ \pmatrix{ 1&0&0&0\\ 0&2&1&0\\ 0&1&0&0\\ 0&0&0&1} \quad \text{...III= 2 III - II}\\ \pmatrix{ 1&0&0&0\\ 0&2&0&0\\ 0&0&-2&0\\ 0&0&0&1} $$ I'm not sure if this is correct

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  • $\begingroup$ The point is that we have to do the same column operations as the row operations immidietly after each row operation. $\endgroup$ – NightRa Apr 5 '14 at 20:37
  • $\begingroup$ The third matrix should be euqal to the second after that operation. $\endgroup$ – NightRa Apr 5 '14 at 21:08
  • $\begingroup$ Are you sure about that? Anyway, if you go through the matrix multiplication $P^TAP$, you do get a diagonal matrix $\endgroup$ – Omnomnomnom Apr 5 '14 at 21:09
  • $\begingroup$ Yeah. You've missed the column operation there. The interesting point is that a symmetric matrix is congruent to every diagonal matrix with the same signed elements in the diagonal. $\endgroup$ – NightRa Apr 5 '14 at 21:12
  • $\begingroup$ @NightRa row operation gets you to $$ \pmatrix{ &&&\\ &1&0&\\ &1&1&\\ &&&\\ } $$ And then the column operation gives you $$ \pmatrix{ &&&\\ &1&0&\\ &0&1&\\ &&&\\ } $$ $\endgroup$ – Omnomnomnom Apr 5 '14 at 21:45
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A fool proof algorithm :)

1) Find the eigenvalues. The eigenvalues are the roots of the polynomial $det(A-\lambda I)$ The eigenvalues are also the values that you will see on the diagonal.

2) Find the eigenvectors, and insert them as columns of a matrix named $P$.

Edit: in your case, since the matrix is symmetric, not only will you find a basis of eigenvectors, you will find an orthonormal basis of eigenvectors.

3) check that $P^{-1}AP = D$ where $D$ is a diagonal matrix with the eigenvalues on the diagonal.

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  • $\begingroup$ Good answer :) . $\endgroup$ – MathMan Jun 20 '14 at 21:21
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Although I'm quite late, I think I've got the solution acording to your algorithm.

As far as I know the algorithm is known as "symmetric Gaussian elimination". It's basically the normal Gaussian elimination for finding the inverse of a matrix, where you record the row operations on the right in the identity matrix. But for every row operation you do the exact same operation for the columns without recording it in the identity matrix and you stop as soon as you have gotten a diagonal matrix.

So here you go (rows in Latin, columns in Arabic):

$$ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&0&1&0 & 0&1&0&0\\ 0&1&0&0 & 0&0&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad (\;A\;\;|\;\;\mathbb{1}\;) $$

$$ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&1&1&0 & 0&1&1&0\\ 0&1&0&0 & 0&0&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{II = II+III}\\ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&1&0 & 0&1&1&0\\ 0&1&0&0 & 0&0&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{2 = 2 + 3}\\ $$

$$ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&1&0 & 0&1&1&0\\ 0&3&1&0 & 0&1&2&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{III = III+II}\\ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&3&0 & 0&1&1&0\\ 0&3&4&0 & 0&1&2&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{3 = 3 + 2}\\ $$

$$ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&3&0 & 0&1&1&0\\ 0&0&-1&0 & 0&-1&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{III = 2*III - 3*II}\\ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&0&0 & 0&1&1&0\\ 0&0&-2&0 & 0&-1&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad \text{3 = 2*3 - 3*2}\\ $$

$$ \left[ \begin{array}{cccc|cccc} 1&0&0&0 & 1&0&0&0\\ 0&2&0&0 & 0&1&1&0\\ 0&0&-2&0 & 0&-1&1&0\\ 0&0&0&1 & 0&0&0&1\\ \end{array} \right] \quad (\;D\;\;|\;\;P\;) $$

Please let me know if I made any errors in the calculation.

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