Equivalently X=RZT (X=RZ* in C).

I think that a polar decomposition can be useful:  https://en.wikipedia.org/wiki/Polar_decomposition

Equivalently X=RZT (X=RZ* in C).

I think that a polar decomposition can be useful:  https://en.wikipedia.org/wiki/Polar_decomposition

Thank you Viera! I will check the polar decomposition.

Thank you Panchatcharam Mariappan! I think that the proposition is true.

The given matrix is non-singular. The question is to find the orthogonal matrix so that multiplying by this orthogonal matrix we obtain the positive definite symmetric matrix. The matrices are with real entries.

The target matrix is symmetric. Otherwise we could use RQ decomposition X=RZ, where X is non-singular, R is upper triangular and Z is orthogonal. XZ^T=R. It is clear that we can multiply both sides with the other orthogonal matrix XZ^TS=RS to obtain in the right side upper triangular matrix with numbers > 0 on the diagonal.

Sorry for asking not very important question but I can not reach the specialist in linear algebra in reality.

Yes. Sorry. I realized that, there is another one available A=P'U which I forgot to notice. Viera is correct.

Dear Maxim, please, explain your problem with answering the following questions:

1. Is your problem to prove the theorem: If  X  is a nonsingular matrix, then there exists an orthogonal matrix  Z  such that   X*Z is a  symmetric positive definite matrix.?

2. Is the theorem already known from some works, and you are lookong for the proof, only?

3. Do you admit complex valued matrices?

Sincerely, Joachim

Dear Joachim,

1. right.

2. yes, I found this proposition in the lecture notes of Arkadii Nemirovski on convex optimization. He metioned this as a known result from Linear Algebra.

3. I'm interested in the real matrices.

With respect,

Maksim

 Dear Maxim, thanks for quick answer. Up to now I have found only the proof for  2 by 2 matrices, which is really simple.but a little too tedious for brief letter. Perhaps some time later I have time to "release" it.  Let us wait forsome more skilled RGaters,who find the proof for the more difficult cases:)

Regards, Joachim

PS Our lettesr were sent almost simultaneously, though my questions 1 - 3 look like repeated unnecessarily. But the System is guilty, not me:) JoD

Dear Joachim,

Thank you for the answer!

At least you talk to me, many don't want to deal with the difficult problems :)

I hope that my questions will not cause problems for the experienced scientists to tell the answers.

With respect,

Maksim

Dear Maksim,

thank you for your question. It pushed me up to revise and complete my knowledge concerning a decomposition of a matrix.

If you would like to have a proof concerning the polar decomposition, I invite you to read the lecture of Travis Schedler in http://math.mit.edu/~trasched/18.700.f11/lect19-article.pdf , more precisely points (5), (6) and (7).

Thank you Viera very much!