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Decomposing a matrix is like how we refactor a codebase
The Story
Today, let’s talk about matrix diagonalization in a way that connects with how we think as developers.
What Is Matrix Diagonalization?
Diagonalization aims to transform a complex matrix into a diagonal one. Why? Because diagonal matrices are simpler to work with — they’re like the “clean code” of the matrix world.
Firstly, let’s talk about decomposing a matrix (the reverse operation of multiplying a matrix, which ‘combines’ 2 transformations into one), decomposing a matrix is like breaking down a giant, messy class into clear, modular components.
Diagonal matrices represent one of the simplest and most fundamental ways to decompose a matrix.
The formula for diagonalization looks like this:
A = P · D · P⁻¹Where:
- A: Your original matrix — the one you want to simplify.
- D: A diagonal matrix containing the eigenvalues of A.
- P: A matrix whose columns are the eigenvectors of A.
Eigenvalues and Eigenvectors
Eigenvalues are like scaling factors, and eigenvectors are special directions. Imagine you’re applying a transformation to a rubber sheet. Most points on the sheet move in unpredictable…
