Diagonalization

Chapter 3 • Section 3-3

"Finding the 'natural' axes of a matrix. 📏"

✨ Eigen-Things

The Definition

An eigenvector of $A$ is a non-zero vector $\vec{x}$ such that $A\vec{x}$ is just a scalar multiple of $\vec{x}$.

$$ A\vec{x} = \lambda \vec{x} $$

$\lambda$ is the eigenvalue.

Visualizer: $A = \begin{bmatrix} 2 & 0 \\ 0 & 0.5 \end{bmatrix}$

Move your mouse to change $\vec{x}$ (blue). See $A\vec{x}$ (red).

Geometric Interpretation

An eigenvector $\vec{v}$ defines a line $L_{\vec{v}}$ through the origin.

This line is invariant under $A$: if $\vec{x}$ is on the line, $A\vec{x}$ is also on the line (just scaled by $\lambda$).

How to Find Them?

Step 1: Characteristic Equation

Solve for $\lambda$:

$$ \det(A - \lambda I) = 0 $$

Step 2: Find Eigenvectors

For each $\lambda$, find the null space of $(A - \lambda I)$:

$$ (A - \lambda I)\vec{x} = \vec{0} $$

🧠 Quick Calc Win $10

Find the eigenvalues of $A = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$.

Hint: For diagonal matrices, it's easy!

Diagonalization

The Theorem

If an $n \times n$ matrix $A$ has $n$ linearly independent eigenvectors, we can write:

$$ A = PDP^{-1} $$
  • $D$ is a diagonal matrix of eigenvalues.
  • $P$ is a matrix with eigenvectors as columns.
  • $D$ is a diagonal matrix of eigenvalues.
  • $P$ is a matrix with eigenvectors as columns.

Why do we care?

It makes computing powers super fast!

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

Computing $D^k$ is trivial (just power the diagonal entries).

Example: Computing $A^{100}$

Let $A = \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}$. We found $P = \begin{bmatrix} 1 & -2 \\ 1 & 1 \end{bmatrix}$ and $D = \begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix}$.

Then $A^{100} = P D^{100} P^{-1}$:

$$ A^{100} = \begin{bmatrix} 1 & -2 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 2^{100} & 0 \\ 0 & 5^{100} \end{bmatrix} \frac{1}{3} \begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix} $$

Much easier than multiplying $A$ by itself 100 times!

🤔 Logic Check Win $20

Can the matrix $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ be diagonalized?