Diagonalizable matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is called diagonalizable or non-defective if it is <a href="/facts/Matrix_similarity/ySevpav1">similar</a> to a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a>. That is, if there exists an <a href="/facts/Invertible_matrix/fPqXk3V8">invertible matrix</a> 
 
 
 
 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 <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
, a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> 
 
 
 
 T
 :
 V
 →
 V
 
 
 {\displaystyle T:V\to V}
 
 is called diagonalizable if there exists an <a href="/facts/Basis_(linear_algebra)/89IPoN6c">ordered basis</a> of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 consisting of <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a> of 
 
 
 
 T
 
 
 {\displaystyle T}
 
. These definitions are equivalent: if 
 
 
 
 T
 
 
 {\displaystyle T}
 
 has a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 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 <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> 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 <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 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 <a href="/facts/Inhomogeneous_dilation/pFm6fvpu">inhomogeneous dilation</a> (or anisotropic scaling). That is, it can <a href="/facts/Scaling_(geometry)/pFm6fvpu">scale</a> 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 <a href="/facts/Defective_matrix/I5uysqAs">defective</a>. It can happen that a matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 with <a href="/facts/Real_number/R02gw5Pb">real</a> entries is defective over the real numbers, meaning that 
 
 
 
 A
 =
 P
 D
 
 P
 
 −
 1
 
 
 
 
 {\displaystyle A=PDP^{-1}}
 
 is impossible for any invertible 
 
 
 
 P
 
 
 {\displaystyle P}
 
 and diagonal 
 
 
 
 D
 
 
 {\displaystyle D}
 
 with real entries, but it is possible with <a href="/facts/Complex_number/2w7mqI7e">complex</a> entries, so that 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is diagonalizable over the complex numbers. For example, this is the case for a generic <a href="/facts/Rotation_matrix/c2K2Ftl8">rotation matrix</a>. 
Many results for diagonalizable matrices hold only over an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a> (such as the complex numbers). In this case, diagonalizable matrices are <a href="/facts/Dense_set/nPT9Yxao">dense</a> in the space of all matrices, which means any defective matrix can be deformed into a diagonalizable matrix by a small <a href="/facts/Perturbation_theory/JWrNgbVm">perturbation</a>; and the <a href="/facts/Jordan%25E2%2580%2593Chevalley_decomposition/w4PN88Xo">Jordan–Chevalley decomposition</a> states that any matrix is uniquely the sum of a diagonalizable matrix and a <a href="/facts/Nilpotent_matrix/qIns3FJJ">nilpotent matrix</a>. Over an algebraically closed field, diagonalizable matrices are equivalent to <a href="/facts/Semi-simplicity/Wz7E1GbT">semi-simple matrices</a>.

Diagonalizable matrix open-in-new

Diagonalizable matrix