real symmetric similar matrix

May 2010
2
0
I have a matrix A which is positive definite and having all real elements. A can have multiple eigenvalues but it is not defective.
I want to get a similar matrix B (i.e., B has the same eigenvalues as A) which is symmetric and real. So precisely I want to construct a nonsingular matrix X such that B = inv(X) A X.
So it will be extremely helpful if anyone can tell me how to get such X or at least give me some reference.

Alternatively:
A method is available to get a symmetric similar matrix for A from another nonsingular symmetric matrix Y if it satisfies A Y = Y A_tr. But such similar matrix obtained from Y will only be real if Y is positive definite. So alternatively, any help in getting a positive definite Y for given A will be extremely useful.
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
i have a matrix a which is positive definite and having all real elements. A can have multiple eigenvalues but it is not defective.
I want to get a similar matrix b (i.e., b has the same eigenvalues as a) which is symmetric and real. So precisely i want to construct a nonsingular matrix x such that b = inv(x) a x.
So it will be extremely helpful if anyone can tell me how to get such x or at least give me some reference.

Alternatively:
A method is available to get a symmetric similar matrix for a from another nonsingular symmetric matrix y if it satisfies a y = y a_tr. But such similar matrix obtained from y will only be real if y is positive definite. So alternatively, any help in getting a positive definite y for given a will be extremely useful.

If we assume A is a symmetric and real, then B will be real; however, not sure if that implies B will be symmetric. If it does imply that B is symmetric, then this is how I would show it.

\(\displaystyle A=A^T\)
\(\displaystyle B=X^{-1}AX\)

\(\displaystyle B^T=(X_1^{-1}AX_1)^T=X_1^TA^T(X_1^{-1})^T=((X_1^T)^{-1})^{-1}A^T(X_1^{-1})^T\)
\(\displaystyle X=(X_1^T)^{-1}\)
\(\displaystyle B^T=X^{-1}A^TX\)
 
May 2010
2
0
Thanks for your reply. The main problem is that the matrix \(\displaystyle A\) is not symmetric. If it is so then there is no need to get any symmetrizer for that. I can work with \(\displaystyle A\) itself.