Diagonalizable matrix
In linear algebra, a square matrix
A is called
diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix
P such that
P−1
AP is a diagonal matrix. If
V is a finite-dimensional vector space, then a linear map
T :
V →
V is called
diagonalizable if there exists an ordered basis of
V with respect to which
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 which is not diagonalizable is called
defective. Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their eigenvalues and eigenvectors are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power. Geometrically, a diagonalizable matrix is an
inhomogeneous dilation — 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.