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Inverse Matrix

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Inverse Matrix

Inverse Matrix

Prerequisites

1

Determinant

What is Inverse Matrix

1. What is Inverse Matrix?

Think geometrically:

  • $A$ → Transformation
  • $X$ → vector in the “before” space
  • $Y$ → vector in the “after” space

So solving:

\[AX = Y \quad \rightarrow \quad X = A^{-1} Y\]

A matrix is transformation from original space to A space. But if we want to retransform from A space to original space, using inverse.

when we transform space, there’re two features, scaling(included in orienetation) and direction.

Scaling

If it can be reformed from determinant.

Direction

If it can be reformed from inverse direction.

2. Depends on determinant.

2-1. $\det(A) \ne 0$

  • Space is NOT collapsed
  • Transformation is invertible
  • Every output vector corresponds to exactly one input vector
\[AX = Y \quad \rightarrow \quad X = A^{-1} Y\]

The result have no squishing, no dimension loss.

Let’s think A matrix is right shear and A inverse matrix is left shear.

Right shear:

\[\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\]

Inverse is left shear:

\[\begin{bmatrix} 1 & -k \\ 0 & 1 \end{bmatrix}\]

Original space is transformed to A space and retransform from A space to original space. The two transform result is same like the first time.

\[A^{-1}A = I\]

This transformation does nothing.

2-2. $\det(A) = 0$

matrix-inverse

  • Space is squished to lower dimension
  • Transformation is not invertible
  • Some vectors collapse together

Multiple input vectors map to the same output vector.

That’s why sometimes there’re no solution or infinitely many solutions.

3. How to calculate inverse matrix.

\[A^{-1}= \frac{1}{\det(A)} \, \operatorname{adj}(A)\]

where: \(\operatorname{adj}(A) = \left[ C_{ij} \right]^T , \quad C_{ij}= (-1)^{i+j} \det(M_{ij})\)

Ex. Numerical Calculation

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}\]

determinant

\[\det(A) = 1 \begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} - 2 \begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} + 3 \begin{vmatrix} 0 & 1 \\ 5 & 6 \end{vmatrix}\]

Cofactor Matrix

\[C = \begin{bmatrix} -24 & 20 & -5 \\ 18 & -15 & 4 \\ 5 & -4 & 1 \end{bmatrix}\]

where:

\[C_{11} = \begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} = 1\cdot0 - 4\cdot6 = -24 , \quad C_{12} = \begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} = 0\cdot0 - 4\cdot5 = -20 , \quad ...\]

Adjugate:

\[\operatorname{adj}(A) = C^T = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}\]
Mathematics, Mathematics - Linear Algebra
Mathematics Mathematics - Linear Algebra
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