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If $D$ is a diagonal matrix, with non-negative eigenvalues, prove that there is a matrix $S$ such that $S^2 = D$

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  • $\begingroup$ I really don't know how to go about proving this because there are no specific values. Can someone please help? $\endgroup$ – user115241 Dec 12 '13 at 1:04
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    $\begingroup$ How would you do it if you knew the values of the diagonal of $D$? $\endgroup$ – user7530 Dec 12 '13 at 1:11
  • $\begingroup$ I don't have the values. $\endgroup$ – user115241 Dec 12 '13 at 1:15
  • $\begingroup$ He's asking you how you would do it if you did.... $\endgroup$ – Don Larynx Dec 12 '13 at 1:25
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Since $D$ is a diagonal matrix, its eigenvalues are on the diagonal, which you know are non-negative. You can show, using matrix multiplication that for a diagonal matrix, $S$, $S^2$ is $S$ with the values on the diagonal squared. Since all the diagonal values of $D$ are non-negative, then there must be a matrix $S$ such that $S^2 = D$.

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  • $\begingroup$ Thank you for your help. What you're saying makes sense, but how would I show that arbitrarily because I am given no values of the matrix. $\endgroup$ – user115241 Dec 12 '13 at 1:14
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    $\begingroup$ @user115241: If $D=Diag(d_1,\ldots, d_k)$, let $S=Diag(\sqrt{d_1},\ldots, \sqrt{d_k})$. $\endgroup$ – Cheerful Parsnip Dec 12 '13 at 1:16
  • $\begingroup$ You can say that $D_{ii} = \lambda_i$, where $D_{ii}$ is the $i,i$ entry of $D$. Then define $S$ such that $S_{ii} = \sqrt{\lambda_i}$ and $0$ everywhere else. $\endgroup$ – Christopher Liu Dec 12 '13 at 1:17
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You seem to be hung up on the fact that "you weren't given specific values". Well, we know that the matrix is a diagonal matrix, so it has to look something like this: $$ D = \pmatrix{ d_{11}&0&\cdots&0\\ 0&d_{22}&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&d_{nn} } $$ Now, we don't know what the values from $d_{11}$ to $d_{nn}$ are, but we know that every other entry of the matrix is zero.

Now, what can we say about the eigenvalues of $D$? As Christopher said, the eigenvalues of $D$ are exactly the diagonal entries $d_{11}$ to $d_{nn}$. Why is this the case? Check the definition of an eigenvalue, and check to see why this has to be true.

Now that we know that $d_{11},\dots,d_{nn}$ are the eigenvalues, we know that they have to be non-negative. Since they are non-negative numbers, they have a non-negative square root. Now, look at the matrix $$ S = \pmatrix{ \sqrt {d_{11}}&0&\cdots&0\\ 0&\sqrt{d_{22}}&\cdots&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots& \sqrt{d_{nn}} } $$ What happens when you multiply $S$ by $S$, using the rules for scalar multiplication? If you can't figure it out right away, try it for the $2\times 2$ and $3 \times 3$ version, and see if you can find the pattern.

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