How can I conformally partition A so that I can perform the following matrix muliplication? $ \mathbf U_1^* \mathbf A \mathbf V_1 $

Question

Let $ \sigma_1 $ be the 2-norm of $\mathbf A$; there exist unit length vectors $\mathbf x_1 \in \mathbb{C}^m,\space \mathbf x_1^*\mathbf x_1 = 1 $ and $ \mathbf y_1 \in \mathbb{C}^n,\space \mathbf y_1^*\mathbf y_1 = 1$ , such that $ \mathbf A \mathbf x_1 = \sigma_1 \mathbf y_1. $ Define the unitary matrices $ \mathbf V_1, \mathbf U_1 $ so that their first column is $ \mathbf x_1, \mathbf y_1 $, respectively:$ \mathbf V_1 = [\mathbf x_1\space \hat{\mathbf V}_1],\space \mathbf U_1 = [\mathbf y_1\space \hat{\mathbf U}_1] $

How can I conformally partition $\mathbf A$ so that I can perform the following matrix muliplication? $$ \mathbf U_1^* \mathbf A \mathbf V_1 $$ Thanks in advance for your time and effort.

Best,

Tri

Answer 1

Recall that for conformal matrix multiplication, the number of columns of the first matches the number of rows of the second. Similarly, in block-matrix multiplication, the partition of the columns of the first must match the partition of the rows of the second.

So, we must patition $A$ into $$ A = \pmatrix{\mathbf a_1^* \\ \hat{\mathbf{ A}}_1} $$