Diagonalization

Diagonalization Basics

Eigenvectors give a matrix its natural axes. Diagonalization exploits this: if a matrix has enough independent eigenvectors to form a basis, the transformation is simply scaling along each axis via a diagonal matrix.

A square matrix $A$ is diagonalizable if it can be written as:

$$A = P D P^{-1}$$

An $n \times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors. A sufficient condition is having $n$ distinct eigenvalues, as their eigenvectors are automatically independent.

Powers and Pitfalls

The true payoff of diagonalization is computational. Raising a matrix to a power becomes trivial using the formula:

$$A^k = P D^k P^{-1}$$

Here, $D^k$ just raises each diagonal entry to the power $k$. This avoids enormous multiplication work when computing $A^{100}$, enabling fast solutions to linear recurrences and differential equations.

Common pitfall: Assuming every square matrix is diagonalizable. You must always check the actual number of independent eigenvectors.