I am studying a proof in which in one of the steps we have a term $$\det(tI+xA+sB)$$ where I is an identity Matrix and A,B are Symmetric Matrices (all matrices are of same order) where as t,x,s are in $R$ , here we multiply this term by $(tI+sB)^{-1}$ and we obtain as resultant the term $$\det(I+(tI+sB)^{-1}Ax)$$ At this point in the proof a comment is made that we can now Symmetrize $(tI+sB)^{-1}A$ and the proof continues.
In this proof the value of determinant is of prime importance so the Symmetrization must preserve eigen values.
I read about two ways of Symmetrizing a Matrix:-
For a give Non-Symmetric Matix M we can obtain a Symmetric matrix S by
- $S= \frac{(M+M^t)}2$.
- If M satisfies properties mentioned here
https://en.wikipedia.orgwiki/Symmetric_matrix#Symmetrizable_matrix under the section Symmetrizable matrix then we can find an Invertible Diagonal matrix D such that $S=DMD^{-1}$
First method will not preserve the eigen values because if it was to then any Non-Symmetric Matrix could be converted to a Symmetric Matrix and hence it must have only real eigen values but then it will imply that all matrices must have only real eigen values which is not true.
To find out if the Second method can be used to Symmetrize $(tI+sB)^{-1}A$ I took two $3*3$ Symmetric matrices and calculated their product and then tried verifying the conditions mentioned in the link but it failed.By this I concluded that product of two Symmetric Matrices need not be Symmetrizable.
I am stuck here, is there some other method of Symmetrization which i must study or is it that i have misunderstood something. Please help.