Diagonalizable matrix

In linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P {\displaystyle P} and a diagonal matrix D {\displaystyle D} such that P − 1 A P = D {\displaystyle P^{-1}AP=D} . This is equivalent to A = P D P − 1 {\displaystyle A=PDP^{-1}} . (Such P {\displaystyle P} , D {\displaystyle D} are not unique.) This property exists for any linear map: for a finite-dimensional vector space V {\displaystyle V} , a linear map T : V → V {\displaystyle T:V\to V} is called diagonalizable if there exists an ordered basis of V {\displaystyle V} consisting of eigenvectors of T {\displaystyle T} . These definitions are equivalent: if T {\displaystyle T} has a matrix representation A = P D P − 1 {\displaystyle A=PDP^{-1}} as above, then the column vectors of P {\displaystyle P} form a basis consisting of eigenvectors of T {\displaystyle T} , and the diagonal entries of D {\displaystyle D} are the corresponding eigenvalues of T {\displaystyle T} ; with respect to this eigenvector basis, T {\displaystyle T} is represented by D {\displaystyle D} . Diagonalization is the process of finding the above P {\displaystyle P} and D {\displaystyle D} and makes many subsequent computations easier. One can raise a diagonal matrix D {\displaystyle D} to a power by simply raising the diagonal entries to that power. The determinant of a diagonal matrix is simply the product of all diagonal entries. Such computations generalize easily to A = P D P − 1 {\displaystyle A=PDP^{-1}} . The geometric transformation represented by a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling). That is, it can scale the space by a different amount in different directions....