Purpose of Inverse matrix

Question

What use is the inverse matrix? I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.

Answer 1

Lots of reasons :

• You CAN use it.

• In certain cases, it is the fastest way (Diagonal Matrices).

• As preconditioners to linear systems.

Consider a system $Ax=b$ which is being solved numerically, if it is possible to find a matrix K which approximates the inverse of A, it is possible to solve $KAx=Kb \implies A'x=b' $ for $x$ much faster.

The theory of preconditioners is very rich and active. The hunt for such Approximate inverses is fairly field specific and is crucial in solving linear systems using modern algorithms such as the Krylov Subspace Methods. In such algorithms, the crucial parameter is the condition number which is the ratio of the maximum and minimum eigenvalues. Smaller the ratio (min. being 1), FASTER is the convergence (Identity matrix will converge in 1 step). So, preconditioners essentially reduce the condition number and in the process reduce the number of iterations (& hence time) required for the algorithms to converge.

Example : $$A=\begin{bmatrix} 0.6551 & 0.4984 & 0.5853\\ 0.1626 & 0.9597 & .2238\\ 0.1190 & 0.3404 & 0.7513\\\end{bmatrix} eig(A) = \begin{bmatrix} 1.3665\\ 0.4732\\ 0.5264\\\end{bmatrix}$$ Use a preconditioner $K=\begin{bmatrix} 1.9000 & -0.5000 & -1.3000\\ -0.3000 & 1.2000 & -0.2000\\ -0.2000 & -0.5000 & 1.6000\\ \end{bmatrix}$

For the system $KAx=Kb \Longleftrightarrow A'x = b'$, $$A' = \begin{bmatrix} 1.0087 & 0.0245 & 0.0235\\ -0.0252 & 0.9341 & -0.0573\\ -0.0219 & -0.0349 & 0.9731\\ \end{bmatrix}; eig(A') = \begin{bmatrix} 0.9169\\ 0.9995 + 0.0035i\\ 0.9995 - 0.0035i\\ \end{bmatrix} $$ How K was obtained is a question that involves delving into the source of the matrix but that's not essential for the motivation of the topic. All that is important is that $K$ attempts to approximate the inverse of the matrix A.