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Question is from Artin's Algebra, p. 263.

If $A$ is the matrix of a symmetric bilinear form, prove or disprove: The eigenvalues of $A$ are independent of the choice of basis.

I suspect the result is true.

Real & Symmetric $\Rightarrow$ Hermitian

Then by Corollary (4.12) the matrices which represent the same hermitian form are $= QAQ^*$, where $Q\in \operatorname{GL}_n(\mathbb{C})$.

Does this mean that all of these matrices are similar? If so I would be done since similar matrices have the same eigenvalues.

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    $\begingroup$ after reading about sylvester's law I suspect such matrices only have the same number of positive, negative and zero eigenvalues $\endgroup$
    – user9352
    Jan 3, 2012 at 2:06
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    $\begingroup$ If a matrix is symmetric, it's Hermitian. $\endgroup$ Jan 3, 2012 at 2:07
  • $\begingroup$ I just wrote up a counter-example in question 96904. Compare the starting and the last matrices' eigenvalues. $\endgroup$
    – user13838
    Jan 14, 2012 at 5:03
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    $\begingroup$ If the question does not mean a similarity transformation by change of basis that is by the way. $\endgroup$
    – user13838
    Jan 14, 2012 at 5:13

2 Answers 2

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I interpret the question as follows:

Let $f$ be a symmetric bilinear form on a finite dimensional vector space over a field $K$ whose characteristic in not two and which has at least four elements, and let $A$ be the matrix of $f$ with respect to some basis $B$ of $V$. Prove or disprove: The eigenvalues of $A$ are independent of the choice of $B$.

We disprove the statement as follows.

Putting $$ V:=K,\quad f(x,y):=xy,\quad B:=\{b\}, $$ we get $A=(b^2)$. In view of the assumptions on $K$, we can choose nonzero $b$ and $b'$ in $K$ so that $b^2\neq b'^2$.

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If you regard those $Q$ as change of basis, then only rigid transformations (isometries/preserving the inner product) preserve eigenvalue. Otherwise if you scale the principal axes like Gaillard does in his answer - $b\neq b’$ - the bilinear form applies a quadratic effect which alters the eigenvalue. Pratically you need $Q$ to be unitary.

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