Editing Diagonalizable matrix (section)

{{Short description|Matrices similar to diagonal matrices}}
{{About|matrix diagonalization in linear algebra||Diagonalization (disambiguation){{!}}Diagonalization}}
{{Use American English|date = April 2019}}

In [[linear algebra]], a [[square matrix]] <math>A</math> is called '''diagonalizable''' or '''non-defective''' if it is [[matrix similarity|similar]] to a [[diagonal matrix]]. That is, if there exists an [[invertible matrix]] <math>P</math> and a diagonal matrix <math>D</math> such that {{nowrap|<math>P^{-1}AP=D</math>}}. This is equivalent to {{nowrap|<math>A = PDP^{-1}</math>.}} (Such {{nowrap|<math>P</math>,}} <math>D</math> are not unique.) This property exists for any linear map: for a [[dimension (vector space)|finite-dimensional]] [[vector space]] {{nowrap|<math>V</math>,}} a [[linear map]] <math>T:V\to V</math> is called '''diagonalizable''' if there exists an [[Basis (linear algebra)#Ordered bases and coordinates|ordered basis]] of <math>V</math> consisting of [[eigenvector]]s of <math>T</math>. These definitions are equivalent: if <math>T</math> has a [[matrix (mathematics)|matrix]] representation <math>A = PDP^{-1}</math> as above, then the column vectors of <math>P</math> form a basis consisting of eigenvectors of {{nowrap|<math>T</math>,}} and the diagonal entries of <math>D</math> are the corresponding [[eigenvalue]]s of {{nowrap|<math>T</math>;}} with respect to this eigenvector basis, <math>T</math> is represented by {{nowrap|<math>D</math>.}} 

'''Diagonalization''' is the process of finding the above <math>P</math> and {{nowrap|<math>D</math>}} and makes many subsequent computations easier. One can raise a diagonal matrix <math>D</math> 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 {{nowrap|<math>A=PDP^{-1}</math>.}} 

The geometric transformation represented by a diagonalizable matrix is an ''[[inhomogeneous dilation]]'' (or ''anisotropic scaling''). That is, it can [[Scaling (geometry)|scale]] the space by a different amount in different directions. The direction of each eigenvector is scaled by a factor given by the corresponding eigenvalue.

A square matrix that is not diagonalizable is called ''[[Defective matrix|defective]]''. It can happen that a matrix <math>A</math> with [[real number|real]] entries is defective over the real numbers, meaning that <math>A = PDP^{-1}</math> is impossible for any invertible <math>P</math> and diagonal <math>D</math> with real entries, but it is possible with [[complex number|complex]] entries, so that <math>A</math> is diagonalizable over the complex numbers. For example, this is the case for a generic [[rotation matrix]].  

Many results for diagonalizable matrices hold only over an [[algebraically closed field]] (such as the complex numbers). In this case, diagonalizable matrices are [[Dense set|dense]] in the space of all matrices, which means any defective matrix can be deformed into a diagonalizable matrix by a small [[Perturbation theory|perturbation]]; and the [[Jordan–Chevalley decomposition]] states that any matrix is uniquely the sum of a diagonalizable matrix and a [[nilpotent matrix]]. Over an algebraically closed field, diagonalizable matrices are equivalent to [[Semi-simplicity#Semi-simple matrices|semi-simple matrices]].