Visual Guide to Diagonalizable 2×2 Matrices

Geometry, eigenvectors and change of basis shown side by side

A 2×2 matrix is diagonalizable when it has two linearly independent eigenvectors. Those eigenvectors form a basis in which the transformation acts by independent scalings.

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Diagonalizable 2×2 matrices

Visual guide to diagonalizable 2 by 2 matrices, comparing matrix classes, geometry, eigenvectors and the factorization A equals C D C inverse.
The chart compares several common classes of diagonalizable 2×2 matrices. Each row connects the matrix, its geometric action, its eigenvectors and eigenvalues, and the corresponding change-of-basis factorization. Click the image to open the full-size version.

Concept

If the columns of C are eigenvectors of A, then C−1 changes coordinates to the eigenvector basis. In that basis, A becomes the diagonal matrix D, whose diagonal entries are the corresponding eigenvalues.

Structure

Diagonalization separates the transformation into three stages: move into the eigenvector basis, scale independently along the eigenvector directions, and return to the original basis. This is the meaning of A = C D C−1.

For a real symmetric matrix, the eigenvectors can be chosen orthonormally. Then C becomes an orthogonal matrix Q, and the factorization simplifies to A = Q D QT.

Key equations

A = C D C⁻¹
C = [x⃗₁ x⃗₂]
D = diag(λ₁, λ₂)
A x⃗ᵢ = λᵢ x⃗ᵢ
A = Q D Qᵀ for real symmetric A

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