Diagonalizable Matrix - Diagonalization

Diagonalization

If a matrix A can be diagonalized, that is,

then:

Writing P as a block matrix of its column vectors

the above equation can be rewritten as

So the column vectors of P are right eigenvectors of A, and the corresponding diagonal entry is the corresponding eigenvalue. The invertibility of P also suggests that the eigenvectors are linearly independent and form the basis of Fn. This is the necessary and sufficient condition for diagonalizability and the canonical approach of diagonalization. The row vectors of P-1 are the left eigenvectors of A.

When the matrix A is a Hermitian matrix (resp. symmetric matrix), eigenvectors of A can be chosen to form an orthonormal basis of Cn (resp. Rn). Under such circumstance P will be a unitary matrix (resp. orthogonal matrix) and P−1 equals the conjugate transpose (resp. transpose) of P.