M. Fiedler and T. L. Markham have an article that may help you:

Article Two results on basic oscillatory matrices

Dear Liming Yuan ,

The following statement is not clear:

I assumed the following:

The nxn square matrices A and B are given. You want to determine two matrices T1 and T2 such that A = T1 B T2 .

If B is invertible, then choose T1 = Identity and T2 = B-1 A.

Or choose T2 = Identity and T1 = AB-1 .

In general, T1 includes n2 unknowns and similarly T1 includes n2 unknowns.

Expand A = T1 B T2 , we obtain n2 equations in 2n2 unknowns.

In general, the result system has infinite number of solutions for T1 and T2 .

Best regards

Dear @ Issam Kaddoura,

Thank you much for your attention. A and B are not necessarily square matrices.

For example, as is shown in the atteched figure, the right matrix is formed by three independent element a b c, and the each element of the left matrix can be described by these three elements. Then, is there a left-multiplication transpose matrix and a right-multiplication transpose matrix that enables the transformation from the right matrix to the left matrix? (These two transpose matrices are made of constants.)

I want to know the existence and unique issues of these two transpose matrices, and under what condition will these two transpose matrices always exist.

Short comment for above example: your so called left-multiplication transpose matrix (T1) shall have 2×1 dimensions, and your right-multiplication transpose matrix (T2) shall have 3×4 dimensions. So, you will still have more unknowns than equations, what in general can lead to the system with infinite number of solutions...

Dear Liming Yuan ,

You can use the direct approach, as explained in my previous answer.

See the attached file.

Best regards

Issam Kaddoura

Thank you very much for sharing your knowledge.

Anton Vrdoljak

Thank you for your answer.