The concept of "reversing" an effect. From restoring original data scaling to decrypting secure messages.
If we have a function f: X → Y mapping x to y, the inverse function f⁻¹: Y → X maps y back to x.
A function that returns its input unchanged: I(x) = x.
In Matrix form, it's the Identity Matrix (I).
| 1 | 0 |
| 0 | 1 |
| 2 |
| 3 |
An inverse exists if and only if the function is Bijective.
Distinct elements map to distinct images. No two inputs give the same output.
Every element in the co-domain is covered. No "leftover" outputs.
We scale data so units don't affect the model (e.g., House Price vs Rooms). After training, we need to interpret predictions, so we use the Inverse Transformation.
Financial data (like incomes) is often Right Skewed. We apply Log to make it Normal for the model.
Sensitive data is encrypted before storage. Decryption is the inverse function applied to retrieve original info.
A single scalar value computed from a square matrix. It tells us the scaling factor of the transformation.
Geometrically, it represents the Area scaling.
If |A| = 0, the matrix is Singular (non-invertible). You cannot divide by zero!
| 4 | 7 |
| 2 | 6 |
| 6 | -7 |
| -2 | 4 |
Normally, when a matrix transforms a vector, it knocks it off its original line.
Eigenvectors (v) are special vectors that stay on their own line during transformation. They only get stretched or compressed.
The factor by which they stretch/compress is the Eigenvalue (λ).