Diagonalize Matrix Calculator

Matrix diagonalization involves factoring a square matrix $A$ into the product $P D P^{-1}$, where $D$ is a diagonal matrix containing the eigenvalues of $A$, and $P$ is an invertible matrix formed by the corresponding linearly independent eigenvectors. The process begins by calculating the characteristic equation $\det(A - \lambda I) = 0$ to find the eigenvalues ($\lambda$). Once the eigenvalues are established, you solve the homogeneous system $(A - \lambda I)v = 0$ for each eigenvalue to find the corresponding eigenvectors ($v$). If an $n \times n$ matrix possesses $n$ linearly independent eigenvectors, it is fully diagonalizable.

Consider a 2x2 matrix $A$ where $a_{11} = 4$, $a_{12} = 1$, $a_{21} = 2$, and $a_{22} = 3$. First, we set up the characteristic polynomial by subtracting $\lambda$ along the main diagonal: $\det(\begin{smallmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{smallmatrix}) = 0$. Expanding this yields $(4-\lambda)(3-\lambda) - (1)(2) = \lambda^2 - 7\lambda + 10 = 0$. Factoring the quadratic gives $(\lambda - 5)(\lambda - 2) = 0$, revealing our eigenvalues: $\lambda_1 = 5$ and $\lambda_2 = 2$. Next, substituting $\lambda_1 = 5$ into $(A - 5I)v = 0$ yields the eigenvector $\left[1, 1\right]^T$. Doing the same for $\lambda_2 = 2$ gives the eigenvector $\left[1, -2\right]^T$. Thus, our diagonal matrix is $D = \operatorname{diag}(5, 2)$ and our transition matrix is $P = \begin{smallmatrix} 1 & 1 \\ 1 & -2 \end{smallmatrix}$.

Always double-check your initial entry values, as a single sign error can drastically alter the resulting eigenvalues and eigenvectors. Remember that not all matrices can be diagonalized; deficient matrices that lack a full set of linearly independent eigenvectors will fail the diagonalization test. When working manually alongside the calculator, always verify your eigenvector columns by performing the matrix multiplication check $AP = PD$.