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## Summary:

- For given matrices ##A##, ##B## and ##C##, solve the equation $$AXB^T = C$$ for ##X## in terms of the LU decompositions of ##A## and ##B##. When are there no solutions?

## Main Question or Discussion Point

I know that every ##m×n## matrix ##D## can be expressed in the form

$$P_1DP_2 = LU$$

where ##P_1## and ##P_2## are permutation matrices, ##L## is a unit lower triangular matrix, and ##U## has the form

$$\begin{bmatrix}

U_1 & U_2 \newline

0 & 0 \newline

\end{bmatrix}$$

where ##U_1## is an invertible upper triangular matrix (i.e. ##U_1## has nonzero diagonal entries) with the same rank as ##D##.

Furthermore, some useful properties of permutation matrices include ##P_1^{-1}=P_1^T##.

Let the LU decompositions of ##A## and ##B## be

$$P_1 A P_2 = L_A U_A, \quad Q_1 B Q_2 = L_B U_B.$$

Then we can write the matrix equation that we are supposed to solve as follows:

$$P_1^T L_A U_A P_2^T X Q_2 U_B^T L_B^T Q_1 = C.$$

My plan was to try writing this equation in the form ##GY=H## for known ##G## and ##H##, and unknown ##Y## (without assuming that ##U_A## and ##U_B## are invertible). This idea leads to

$$U_A P_2^T X Q_2 U_B^T = L_A^{-1} P_1 C Q_1^T (L_B^T)^{-1} .$$

Then we can read off

$$G=U_A, \quad H = L_A^{-1} P_1 C Q_1^T (L_B^T)^{-1}, \quad Y=P_2^T X Q_2 U_B^T .$$

Am I on the right track? Where do I go from here?

$$P_1DP_2 = LU$$

where ##P_1## and ##P_2## are permutation matrices, ##L## is a unit lower triangular matrix, and ##U## has the form

$$\begin{bmatrix}

U_1 & U_2 \newline

0 & 0 \newline

\end{bmatrix}$$

where ##U_1## is an invertible upper triangular matrix (i.e. ##U_1## has nonzero diagonal entries) with the same rank as ##D##.

Furthermore, some useful properties of permutation matrices include ##P_1^{-1}=P_1^T##.

Let the LU decompositions of ##A## and ##B## be

$$P_1 A P_2 = L_A U_A, \quad Q_1 B Q_2 = L_B U_B.$$

Then we can write the matrix equation that we are supposed to solve as follows:

$$P_1^T L_A U_A P_2^T X Q_2 U_B^T L_B^T Q_1 = C.$$

My plan was to try writing this equation in the form ##GY=H## for known ##G## and ##H##, and unknown ##Y## (without assuming that ##U_A## and ##U_B## are invertible). This idea leads to

$$U_A P_2^T X Q_2 U_B^T = L_A^{-1} P_1 C Q_1^T (L_B^T)^{-1} .$$

Then we can read off

$$G=U_A, \quad H = L_A^{-1} P_1 C Q_1^T (L_B^T)^{-1}, \quad Y=P_2^T X Q_2 U_B^T .$$

Am I on the right track? Where do I go from here?