Diagonalize a 2×2 real matrix using eigenvalues and eigenvectors. Find the matrices P, D, and P⁻¹ when diagonalization is possible.
Real 2×2 matrices only | Eigenvalue/eigenvector method
| Matrix | Notes |
|---|---|
| [[4, 1], [0, 2]] | Distinct real eigenvalues, so diagonalizable |
| [[5, 2], [0, 1]] | Upper triangular matrix with easy eigenvalues |
| [[3, 0], [0, 3]] | Already diagonal |
| [[2, 1], [0, 2]] | Repeated eigenvalue, not diagonalizable |
Diagonalizing a matrix means rewriting it in the form A = P D P⁻¹, where D is a diagonal matrix and P is made from eigenvectors of the original matrix.
This is useful because diagonal matrices are much easier to work with. Powers of matrices, differential equations, dynamical systems, and many linear algebra problems become simpler after diagonalization.
A matrix can be diagonalized when it has enough linearly independent eigenvectors. For real 2×2 matrices, distinct real eigenvalues guarantee diagonalizability.
Solve the characteristic equation det(A − λI) = 0. For a 2×2 matrix, this gives a quadratic equation in λ.
For each eigenvalue λ, solve (A − λI)v = 0. Each nonzero solution gives an eigenvector.
Put the eigenvectors as columns in the matrix P.
Place the corresponding eigenvalues on the diagonal of D in the same order as their eigenvectors appear in P.
If P is invertible, then the matrix is diagonalized successfully.
Example Matrix
Because the matrix has two distinct real eigenvalues, it is diagonalizable.
A matrix is diagonalizable when it can be written as A = P D P⁻¹, where D is diagonal and P is invertible.
Every square matrix has eigenvalues over the complex numbers, but not every real matrix has real eigenvalues.
Yes. A matrix may have repeated eigenvalues without having enough independent eigenvectors.
It makes matrix powers, linear systems, differential equations, and many theoretical calculations easier.
No. This version is for real 2×2 matrices and reports when the eigenvalues are complex.
Then the calculator still works, and the diagonal matrix is simply D with a straightforward choice of P.
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