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I am wondering why

the rank of a symmetric matrix equals its number of nonzero eigenvalues.

I have tried showing it like this:

A symmetrix matrix A can be written:

$$A=PDP^T$$, where P is an orthogonal matrix.

It is not difficult to see that for a vector x: $PDP^Tx=0 \leftrightarrow DP^Tx=0$, since P is invertible.

So what we need to show is that dimension of the nullspace of $DP^T$ equals the number of eigenvalues with value zero.

Do you see how to do this?

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2 Answers 2

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The rank of $A$ is equal to the rank of $D$ and it is clear that the rank of $D$ equals the number of nonzero eigenvalues (which are the same eigenvalues as those of $A$).

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  • $\begingroup$ But how do we know that the rank of A is equal to the rank of D? $\endgroup$
    – user394334
    Jul 21, 2019 at 17:37
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    $\begingroup$ Because two similar matrices have the same rank. $\endgroup$ Jul 21, 2019 at 17:38
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    $\begingroup$ @user394334 Because multiplying with an invertible matrix doesn't affect rank. You've already seen that that's true for multiplying with $P$ from the left. It's not very difficult to show that it also holds for multiplying with $P^T$ from the right. $\endgroup$
    – Arthur
    Jul 21, 2019 at 17:43
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More precisely, I would say that the rank of a symmetric matrix is equal to the sum of the geometric multiplicities of its nonzero eigenvalues. For example, If $A$ has two nonzero eigenvalues, say $2$ and $3$, with geometric multiplicities $1$ and $2$ respectively, then the rank of $A$ is $1+2 = 3$.

In this example, $2$ and $3$ would appear once and twice in $D$ respectively. So the number of repetitions of nonzero eigenvalues in $D$ needs to be considered

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