Eigenbasis

An eigenbasis for a linear transformation T on a vector space V is a basis consisting entirely of eigenvectors of T. In a finite-dimensional space, if such a basis exists, the matrix of T relative to that basis is diagonal, with the eigenvalues on the diagonal. Conversely, diagonalizability means the space has an eigenbasis; if not, no eigenbasis exists.

To determine whether an eigenbasis exists, one looks at the eigenvalues and their eigenspaces. The vector space V is the direct sum of the eigenspaces iff T is diagonalizable. Equivalently, the sum of the geometric multiplicities of all eigenvalues equals the dimension of V. The geometric multiplicity of an eigenvalue λ is the dimension of the null space of (T − λI); its algebraic multiplicity is its multiplicity as a root of the characteristic polynomial.

Finding an eigenbasis involves solving (T − λI)v = 0 for each eigenvalue λ, choosing a basis for each eigenvector space, and then combining bases from the distinct eigenspaces. If this combined set has n linearly independent vectors (n = dimension of V), it forms an eigenbasis and T is diagonalizable.

Special cases: over the real numbers, a matrix may lack a real eigenbasis, but over the complex numbers every diagonalizable matrix has a complex eigenbasis. For real symmetric (Hermitian) matrices, there exists an orthonormal eigenbasis, by the spectral theorem, allowing a diagonalization by an orthogonal (or unitary) change of basis.