My book says this in an answer to one of the problems (where we're asked to prove that a specific symmetric matrix is diagonalizable)

I know that if there are n distinct eigenvalues of an nxn matrix, then it is diagonalizable...But why would it be true that if the eigenvalues are real, then the matrix is diagonalizable? Couldn't you have an eigenvalue with a multiplicity > 1 that has only one eigenvector, for example, and then not have n linearly independent eigenvectors and not be able to diagonalize the matrix?


The problem in my book is:

Prove that the symmetric matrix is diagonalizable \begin{bmatrix}0&0&a\\0&a&0\\a&0&0\end{bmatrix}

And they solve it here: http://calcchat.com/book/Elementary-Linear-Algebra-7e/7/3/7/

I'm confused on where they say, "Since the eigenvalues are real, A is diagonalizable." I don't know how they draw that conclusion. I already know that it's TRUE that symmetric matrices are diagonalizable and have real eigenvalues, but we're supposed to PROVE this is diagonalizable, not just use what we already know.

  • $\begingroup$ Spoiler: all symmetric matrices are diagonalizable $\endgroup$ – Exodd Mar 12 '17 at 23:24
  • $\begingroup$ I know. This whole section is on that. They want us to prove it. I can't just answer the problem by saying, "this symmetric matrix is diagonalizable because symmetric matrices are diagonalizable." $\endgroup$ – dagny Mar 12 '17 at 23:26
  • $\begingroup$ @dagny You should tell us exactly what they are asking you to show: one could answer this question in many levels of abstraction, from "all symmetric matrices are diagonalizable" to explicitly providing a diagonalization of your "specific symmetric matrix." $\endgroup$ – angryavian Mar 12 '17 at 23:30
  • $\begingroup$ @angryavian Okay, I edited it $\endgroup$ – dagny Mar 12 '17 at 23:36

It doesn't make much sense to compute the eigenvalues without their multiplicity and then say "since the eigenvalues are real, the matrix is diagonalizable" because it doesn't hold in general. It does hold for symmetric matrices but symmetric matrices have real eigenvalues and are diagonalizable so there is no need to calculate anything to deduce your matrix is diagonalizable.

You can say that $A$ is diagonalizable because it is real and symmetric without calculating anything. Alternatively, you can compute the eigenvalues (they will be real) and then compute the geometric multiplicity of each eigenvalue and then conclude $A$ is diagonalizable. It seems that whoever wrote the solution mixed both approaches.

  • $\begingroup$ So I'm correct in thinking the book's answer doesn't really make sense? Also, I don't know what geometric multiplicity is, but am I correct in thinking that, rather than just saying it has real eigenvalues and is therefore diagonalizable, you would need to first figure out how many linearly independent eigenvectors the matrix has before concluding that it's diagonalizable? (provided we ignore the fact that we already know symmetric matrices are diagonalizable and have n eigenvectors) $\endgroup$ – dagny Mar 13 '17 at 0:10
  • $\begingroup$ @dagny: You are correct about both points. The geometric multiplicity of an eigenvalues is just the number of linearly independent eigenvectors associated to this eigenvalue. For the matrix to be diagonalizable, you need "enough" eigenvectors. In your case, there are two eigenvalues so one of them will have to contribute two linearly independent eigenvectors (the geometric multiplicity of this eigenvalue will be two) and the other will contribute only one. $\endgroup$ – levap Mar 13 '17 at 0:15

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