I have a quadratic form, say: $A^2+B^2+4AB$.

If I write it in matrix form $V^TQV$ where symmetric $Q=\left[ {\begin{array}{cc} 1 & 2 \\ 2 & 1 \\ \end{array} } \right]$ everything goes fine with diagonalization.

Conversely, if I try to write it as $V^TQ'V$ where non-symmetric $Q'=\left[ {\begin{array}{cc} 1 & 3 \\ 1 & 1 \\ \end{array} } \right]$, I get different eigenvectors that DO NOT diagonalize the form: i.e. when expressing $A$ and $B$ as function of $A'$ and $B'$, the off-diagonal terms do not go away.

What I am asking is the following: since the expression $V^TQ'V$ seems to work perfectly fine in reproducing the form, why I cannot use it for diagonalization?

What is the profound cause of the fact that I need to use a symmetric matrix for diagonalizing a quadratic form?

which are the feature of symmetric matrices that allow diagonalization of quadratic forms?


The reason to prefer a symmetric matrix over an asymmetric matrix is that the eigenvalues of a symmetric matrix are real. This is no longer the case for asymmetric matrices.

Symmetric matrices are characterized by the fact that not only are their eigenvalues all real, but they can be written as $S=ODO^T$ where $O$ is orthogonal (with real entries) and $D$ is a diagonal matrix of real eigenvalues. Having such a representation is equivalent to being a symmetric matrix.

  • $\begingroup$ In my example also the eigenevalues of $Q'$ are real, though. There should be more than this, I suspect. $\endgroup$ – Arnaldo Maccarone Mar 26 '17 at 7:50
  • $\begingroup$ Your question is subjective. It appears what you are trying to ask is for a set of properties that characterize symmetric matrices in a way that is more natural in the context of quadratic forms. Is this correct? (In general, as this is a math site and not a philosophy site, it is better to ask precise mathematical questions.) $\endgroup$ – pre-kidney Mar 26 '17 at 8:01
  • $\begingroup$ Correct, thank you!. Rephrasing in a more precise way: which are the feature of symmetric matrices that allow diagonalization of quadratic forms? (also editing the main question) $\endgroup$ – Arnaldo Maccarone Mar 26 '17 at 8:13

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