Eigenvalues and Eigenvectors

Diagonalization of a 2×2 Matrix

Form P (eigenvector columns) and D (diagonal eigenvalue matrix). Verify the diagonalization A = PDP⁻¹ by checking AP = PD entry by entry.

Example

Build P and D and verify AP equals PD.

highlighted = computed this step

Step 1 — Set up P and D

Set up A, P, and D for diagonalization.

A=[4123],P=[11−21],D=[2005]A=\begin{bmatrix}4&1\\2&3\end{bmatrix},\quad P=\begin{bmatrix}1&1\\-2&1\end{bmatrix},\quad D=\begin{bmatrix}2&0\\0&5\end{bmatrix}

Step 2 — Compute AP

Multiply A by P.

AP=[25-45]AP=\begin{bmatrix}\hl{2}&\hl{5}\\\hl{-4}&\hl{5}\end{bmatrix}

Step 3 — Compute PD

Multiply P by D.

PD=[25-45]PD=\begin{bmatrix}\hl{2}&\hl{5}\\\hl{-4}&\hl{5}\end{bmatrix}

Step 4 — Verify diagonalization

Verify AP = PD, so A is diagonalized.

AP = PD\text{AP = PD}
diagonalization A matrix A is diagonalizable if A = PDP⁻¹ where P has eigenvectors as columns and D is diagonal with the corresponding eigenvalues. Equivalent condition: AP = PD.