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I'm a chemistry major and I haven't taken much math, but this came up in a discussion of quantum chemistry and my professor said (not very confidently) that if a matrix is diagonalizable, then you should be able to diagonalize it to the identity matrix. I suspect this is true for symmetrical matrices, but not all matrices. Is that correct?

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I'm not sure what he means by "diagonalize" it to the identity matrix it. If you mean that if for some matrix $A$ you have an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$, then yes. But we need not have $D = \alpha I$ for some scalar $\alpha \in \mathbb{C}$. In other words, $\begin{pmatrix} 1 & 0 \\ 0 & 2\end{pmatrix}$ is already diagonal, but cannot be expressed as a multiple of the identity. The fact about symmetric matrices is that if $A = A^T$ then $P$ can be taken so that $P^T = P^{-1}$ with $D$ real. – snarski Jan 30 at 5:19
yes, that's what he meant. Thanks! – twilightisnotaprincess Jan 30 at 5:21
Anyone who makes such a statement has no idea of what diagonalisation is, or at least a very different idea than everybody else in the scientific community. – Marc van Leeuwen Jan 30 at 5:23

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up vote 7 down vote accepted

No. If $PAP^{-1} = I$ where $I$ is the identity then $A = P^{-1}IP = P^{-1}P = I$. So in fact only the identity matrix can be diagonalized to the identity matrix.

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up vote 4 down vote

Take the $0$ $n\times n$ matrix. It's already diagonal (and symmetrical) but certainly can't be diagonalized to the identity matrix.

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Hm... but what about a nxn matrix with non-zero elements? – twilightisnotaprincess Jan 30 at 5:22
There are many reasons why an arbitrary diagonalize matrix need not be diagonalizable to the identity. One is the reason given by @Jim above that shows that the only matrix diagonalizable to the identity is the identity itself. – Ittay Weiss Jan 30 at 5:35
maybe I should restate my question: can a symmetric matrix be diagonalized into a matrix D=αI where α is a scalar. – twilightisnotaprincess Jan 30 at 5:39
the same argument as above will show that the only matrix diagonalizable to $\alpha I$ is $\alpha I$. – Ittay Weiss Jan 30 at 5:45
I see... thanks :) – twilightisnotaprincess Jan 30 at 5:53
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up vote 2 down vote

The usual meaning of "diagonalization" is diagonalization by similarity transform, which takes the form of $PAP^{-1}=D$. As the others have shown, the only matrix that can be diagonalized into the identity matrix is the identity matrix itself.

Yet, depending on the context, your professor may refer to diagonalization by congruence, which takes the form of $P^\ast AP=D$. If he implicitly assumes that $A$ is positive definite, then his assertion is true: $P^\ast AP=D\,\Rightarrow\,(PD^{-1/2})^\ast A(PD^{-1/2})=I$. However, since I know absolutely nothing about quantum chemistry, I can't say if he really meant this.

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