Can every symmetric matrix be a matrix of inertia?

[sfn17]

Hello,

I am often designing math exams for students of engineering.

What I ask is the following:

Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia?

Possibly, there are secret connections between the off-diagonal elements (if not zero) which I should have in mind...

I am not quite sure that the off-diagonal elements can be negative. Is it possible to have negative entries even on the diagonal line?

sfn17

 

[anuttarasammyak]

We have mathematical theorem that any real number symmetric matrix, with no condition on signs of its eigenvalues, can be diagonarized by coordinate transformation via orthogonal matrices.