Two symmetric matrices with same characteristic polynomial are congruent.
I know that the above statement is false, But I can't understand why it's false.
Any help will be appreciated.
2026-03-25 12:24:16.1774441456
Two symmetric matrices with same characteristic polynomial are congruent.
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Hint If $A,B$ are symmetric and real, they are diagonalizable. Moreover, since they have the same characteristic polynomial, they can be diagonalized to the same diagonal matrix.
Use this to prove they are congruent.
Note: For complex, the canonical example of a non-diagonalizable matrix is $$\begin{bmatrix} 1& i \\-i &-1 \end{bmatrix} $$
This matrix has both eigenvalues equal to $0$, which means is not diagonalizable, and hence not congruent with its diagonal.