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.