Linear Algebra (Transformation)

Question

I want to calculate $\hat{X}$ from a simple matrix equation, \begin{equation} \hat{A}\hat{X} = \hat{B} \end{equation} in which $\hat{A}$, $\hat{B}$, and $\hat{X}$ are all square $N\times N$ matrices. However, the matrices are very large and solving for $\hat{X}$ is practically impossible.

Considering that in the end, I want to use $\hat{X}$ to get $\vec{u} = \vec{P}\hat{X}$, is there anyway to directly solve for $\vec{u}$? ($\vec{P}$ and $\vec{u}$ are both $1\times N$ vectors.)

Answer 1

You want to find $$ P A^{-1} B $$ You can do this by first finding $$ Q = P A^{-1} $$ and then finding $QB$.

Finding $Q$ amounts to solving $$ QA = P $$ or, equivalently $$ A^t Q^t = P^t. $$

Solving for $Q^t$ in this situation using a linear-equation solver is generally much faster than inverting $A$.

Approximate matlab code (untested):

function z = solvenoei(A, B, p)
% A and B are n x n; p is 1 x n. 

qt = A' \ p'; % find q^t
z = qt' * B;