A matrix is diagonalizable if it can be transformed into a diagonal matrix via a similarity transformation. More formally, an n×nn\times nn×n matrix AAA is diagonalizable if there exists an invertible matrix PPP and a diagonal matrix DDD such that
A=PDP−1A=PDP^{-1}A=PDP−1
Key conditions for a matrix to be diagonalizable include:
- The matrix must have nnn linearly independent eigenvectors. This means you can form PPP entirely out of eigenvectors of AAA.
- Equivalently, the sum of the dimensions of the eigenspaces (the geometric multiplicities of the eigenvalues) must be equal to nnn.
- For each eigenvalue, its geometric multiplicity (number of linearly independent eigenvectors for that eigenvalue) must equal its algebraic multiplicity (multiplicity as a root of the characteristic polynomial). If this is not true, the eigenvalue is considered defective, and the matrix is not diagonalizable.
- A sufficient (but not necessary) condition: if the matrix has nnn distinct eigenvalues, it is guaranteed to be diagonalizable.
- Special cases: symmetric matrices (in the real case) and Hermitian matrices (in the complex case) are always diagonalizable.
In summary, a matrix is diagonalizable if and only if its eigenvalues provide enough linearly independent eigenvectors to form a basis for the vector space. If any eigenvalue is defective (fewer eigenvectors than its multiplicity), the matrix is not diagonalizable.
