Inverse Matrices

Definition

• A matrix that, when multiplied with the original matrix, results in the identity matrix.
• Notation: $A^{-1}$.

Calculating the Inverse of a Matrix

1. Conditions for Inversibility:
   • Matrix must be square ($n\times n$).
   • Determinant $\ne 0$.
2. Method:

$$\begin{array}{rl}& \text{Given a 2x2 matrix }A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),\\ & \text{The inverse of }A\text{ is calculated as:}\\ & A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right),\\ & \text{provided that }ad-bc\ne 0.\end{array}$$

4. Example:

$$\begin{array}{rl}& \text{Consider matrix }A=\left(\begin{array}{cc}4& 7\\ 2& 6\end{array}\right),\\ & \text{The determinant of }A\text{ is calculated as: }\\ & =(4)(6)-(7)(2)=24-14=10,\\ & \text{Therefore, the inverse of }A\text{ is:}\\ & A^{-1}=\frac{1}{10}\left(\begin{array}{cc}6& -7\\ -2& 4\end{array}\right).\end{array}$$

For larger matrices, use Gaussian Elimination

• todo: example here

Practical Use of Inverse Transformations

• In Unity, objects exist in world space, but the camera views them in camera space. Converting coordinates between these spaces is crucial for accurate interaction, like selecting objects with the mouse.

Example: Object Selection with Mouse Picking

• High-Level Unity Abstraction:
  • Unity provides functions to easily convert screen space (mouse position) to world space.
  • Code Example (High-Level Unity Function):

Ray ray = Camera.main.ScreenPointToRay(Input.mousePosition);
RaycastHit hit;
if (Physics.Raycast(ray, out hit)) 
{
    // Object hit by raycast
}

Under-the-Hood Process:

• Internally, Unity uses inverse transformations to convert the screen space position to world space.
• Code Example (Manual Inverse Transformation):

// Getting the inverse of the camera's projection matrix
Matrix4x4 invProjMat = Camera.main.projectionMatrix.inverse;
 
// Converting mouse position to viewport space
Vector3 mousePosition = Input.mousePosition;
Vector3 viewportPoint = Camera.main.ScreenToViewportPoint(mousePosition);
 
// Transforming to world space
Vector3 worldSpaceRay = invProjMat.MultiplyPoint(viewportPoint);
 
// Creating a ray from camera position in the direction of worldSpaceRay
Ray ray = new Ray(Camera.main.transform.position, worldSpaceRay);
RaycastHit hit;
if (Physics.Raycast(ray, out hit)) 
{
    // Object hit by raycast
}