Matrix Inversion

• categories: Linear algebra, Operation

Definition
The inverse of a square matrix $A\in {\mathbb{R}}^{n\times n}$ is a matrix $A^{-1}$ such that:
$A{A}^{-1}=A^{-1}A=I,$ where $I$ is the identity matrix of size $n\times n$.

A matrix is invertible (or nonsingular) if and only if it is square and its Determinant is nonzero, i.e., $\det (A)\ne 0$.

Intuition
Matrix inversion generalizes division for matrices. If $A$ represents a linear transformation, $A^{-1}$ reverses that transformation.

Key Properties

1. Uniqueness:

   • If $A$ is invertible, its inverse $A^{-1}$ is unique.

2. Inversion of a Product:
   $(AB{)}^{-1}=B^{-1}{A}^{-1},$
   where $A$ and $B$ are invertible.

3. Inversion of a Transpose:
   $(A^T{)}^{-1}=(A^{-1}{)}^T.$

4. Inversion of a Scalar Multiple:
   $(\alpha A{)}^{-1}=\frac{1}{\alpha }{A}^{-1},\alpha \ne 0.$

5. Invertibility and Determinants:
   $\det (A^{-1})=\frac{1}{\det (A)}.$

6. Block Matrices:
   Invertibility of block matrices follows specific conditions. For example, if $A$ and $D-C{A}^{-1}B$ are invertible in:
   $\left[\begin{array}{cc}A& B\\ C& D\end{array}\right],$
   the entire matrix can be inverted using Schur complements.

Methods to Compute $A^{-1}$

1. Analytical Formula (2x2 Matrix):
   For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$:
   $A^{-1}=\frac{1}{\det (A)}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],\det (A)=ad-bc.$

2. Gaussian Elimination:

   • Augment $A$ with $I$: $[A|I]$.
   • Apply row reduction to transform $[A|I]\to [I|A^{-1}]$.

3. Lower–Upper Factorization (LU):

   • Use $A=LU$ (if $A$ is invertible and $LU$ exists), then solve $AX=I$ via:
     • $LY=I$ (forward substitution),
     • $UX=Y$ (backward substitution).

4. Adjugate Method:
   $A^{-1}=\frac{1}{\det (A)}\text{adj}(A),$
   where $\text{adj}(A)$ is the adjugate (transpose of cofactor matrix).

5. Numerical Methods:

   • Iterative methods (e.g., Newton’s method) are used for large matrices.

Applications

1. Solving Linear Systems:

   • $Ax=b$ is solved as $x=A^{-1}b$ if $A^{-1}$ is precomputed.

2. Control Theory:

   • Matrix inverses are crucial in state-space models and transfer functions.

3. Statistical Models:

   • Used in covariance matrix computations and generalized least squares.

4. Signal Processing:

   • Inverse filtering relies on matrix inversions to restore original signals.

Examples

1. For $A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$:
   $\det (A)=2\cdot 4-3\cdot 1=5,$
   $A^{-1}=\frac{1}{5}\left[\begin{array}{cc}4& -3\\ -1& 2\end{array}\right].$

2. For $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right]$, use Gaussian elimination to compute $A^{-1}$.