Diagonalizable matrix

In linear algebra, a square matrix A {displaystyle A} is called diagonalizable or nondefective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P {displaystyle P} such that P − 1 A P {displaystyle P^{-1}AP} is a diagonal matrix. If V {displaystyle V} is a finite-dimensional vector space, then a linear map T : V ↦ V {displaystyle T:Vmapsto V} is called diagonalizable if there exists an ordered basis of V {displaystyle V} with respect to which T {displaystyle T} is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective. Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle; once their eigenvalues and eigenvectors are known, one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power, and the determinant of a diagonal matrix is simply the product of all diagonal entries. Geometrically, a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling) — it scales the space, as does a homogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis (diagonal entries). A square n × n {displaystyle n imes n} matrix A {displaystyle A} over a field F {displaystyle F} is called diagonalizable or nondefective if there exists an invertible matrix P {displaystyle P} such that P − 1 A P {displaystyle P^{-1}AP} is a diagonal matrix. Formally,