Can every symmetric matrix be a matrix of inertia?

AI Thread Summary
Any real 3x3 symmetric matrix can be diagonalized using orthogonal matrices, regardless of the signs of its eigenvalues. However, for a matrix to represent a realistic matrix of inertia, it must have positive eigenvalues, which implies that the diagonal elements should also be positive. Off-diagonal elements can influence the matrix's properties, but they do not necessarily have to be positive. Negative entries are generally not suitable for a matrix of inertia, particularly on the diagonal. Therefore, while any symmetric matrix can be diagonalized, not all can serve as a valid matrix of inertia.
sfn17
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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
 
Engineering news on Phys.org
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.
 

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