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

In [[linear algebra]], a [[square matrix]] <math>A</math> is called '''diagonalizable''' or '''nondefective''' if it is [[similar (linear algebra)|similar]] to a [[diagonal matrix]], i.e., if there exists an [[invertible matrix]] <math>P</math>  such that <math>P^{-1}AP</math> is a diagonal matrix. If <math>V</math> is a finite-[[dimension (linear algebra)|dimension]]al [[vector space]], then a [[linear map]] <math>T : V \mapsto V</math> is called '''diagonalizable''' if there exists an [[Basis (linear algebra)#Ordered bases and coordinates|ordered basis]] of <math>V</math> with respect to which <math>T</math> is represented by a diagonal matrix. '''Diagonalization''' is the process of finding a corresponding [[diagonal matrix]] for a diagonalizable matrix or linear map.<ref name="HornJohnson">{{cite book |title=Matrix Analysis, second edition |first1=Roger A. |last1=Horn |first2=Charles R. |last2=Johnson |isbn=9780521839402 |publisher=Cambridge University Press|year=2013}}</ref> A square matrix that is not diagonalizable is called ''[[Defective matrix|defective]].''

Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle; once [[Transformation matrix#Eigenbasis and diagonal matrix|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'')&nbsp;— it [[Scaling (geometry)|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).

== Definition ==
A square <math>n \times n</math> matrix <math>A</math> over a [[field (mathematics)|field]] <math>F</math> is called '''diagonalizable''' or '''nondefective''' if there exists an invertible matrix <math>P</math> such that <math>P^{-1}AP</math> is a diagonal matrix. Formally,

{{Equation box 1
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|equation = <math>A \in F^{n \times n} \text{ diagonalizable} \iff \exists \, P,P^{-1} \in F^{n \times n} : \; P^{-1}AP \text{ diagonal}</math>
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== Characterization ==
The fundamental fact about diagonalizable maps and matrices is expressed by the following:

* An <math>n \times n</math> matrix <math>A</math> over a field <math>F</math> is diagonalizable [[if and only if]] the sum of the [[dimension (linear algebra)|dimension]]s of its eigenspaces is equal to <math>n</math>, which is the case if and only if there exists a [[basis (linear algebra)|basis]] of <math>F^n</math> consisting of eigenvectors of <math>A</math>. If such a basis has been found, one can form the matrix <math>P</math> having these [[basis vectors]] as columns, and <math>P^{-1}AP</math> will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of <math>A</math>. The matrix <math>P</math> is also known as a [[modal matrix]] for <math>A</math>.
* A linear map <math>T : V \mapsto V</math> is diagonalizable if and only if the sum of the [[dimension (linear algebra)|dimension]]s of its eigenspaces is equal to <math>\operatorname{dim}(V)</math>, which is the case if and only if there exists a basis of <math>V</math> consisting of eigenvectors of <math>T</math>. With respect to such a basis, <math>T</math> will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of <math>T</math>.

Another characterization: A matrix or linear map is diagonalizable over the field <math>F</math> if and only if its [[minimal polynomial (linear algebra)|minimal polynomial]] is a product of distinct linear factors over <math>F</math>. (Put in another way, a matrix is diagonalizable if and only if all of its [[elementary divisor]]s are linear.)

The following sufficient (but not necessary) condition is often useful.
{{bulleted list
| An <math>n \times n</math> matrix <math>A</math> is diagonalizable over the field <math>F</math> if it has <math>n</math> distinct eigenvalues in <math>F</math>, i.e. if its [[characteristic polynomial]] has <math>n</math> distinct roots in <math>F</math>; however, the converse may be false. Consider
: <math>\begin{bmatrix} -1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{bmatrix},</math>

which has eigenvalues 1, 2, 2 (not all distinct) and is diagonalizable with diagonal form ([[similar (linear algebra)|similar]] to <math>A</math>)
: <math>\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}</math>

and [[change of basis|change of basis matrix]] <math>P</math>
: <math>\begin{bmatrix} 1 & 1 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & 3 \end{bmatrix}.</math>

The converse fails when <math>A</math> has an eigenspace of dimension higher than 1. In this example, the eigenspace of <math>A</math> associated with the eigenvalue 2 has dimension 2.
| A linear map <math>T : V \mapsto V</math> with <math>n = \operatorname{dim}(V)</math> is diagonalizable if it has <math>n</math> distinct eigenvalues, i.e. if its characteristic polynomial has <math>n</math> distinct roots in <math>F</math>.
}}

Let <math>A</math> be a matrix over <math>F</math>. If <math>A</math> is diagonalizable, then so is any power of it. Conversely, if <math>A</math> is invertible, <math>F</math> is algebraically closed, and <math>A^n</math> is diagonalizable for some <math>n</math> that is not an integer multiple of the characteristic of <math>F</math>, then <math>A</math> is diagonalizable. Proof: If <math>A^n</math> is diagonalizable, then <math>A</math> is annihilated by some polynomial <math>\left(x^n - \lambda_1\right) \cdots \left(x^n - \lambda_k\right)</math>, which has no multiple root (since <math>\lambda_j \ne 0</math>) and is divided by the minimal polynomial of <math>A</math>.

As a rule of thumb, over <math>\mathbb{C}</math> almost every matrix is diagonalizable. More precisely: the set of complex <math>n \times n</math> matrices that are ''not'' diagonalizable over <math>\mathbb{C}</math>, considered as a [[subset]] of <math>\mathbb{C}^{n \times n}</math>, has [[Lebesgue measure]] zero. One can also say that the diagonalizable matrices form a dense subset with respect to the [[Zariski topology]]: the complement lies inside the set where the [[discriminant]] of the characteristic polynomial vanishes, which is a [[hypersurface]]. From that follows also density in the usual (''strong'') topology given by a [[norm (mathematics)|norm]]. The same is not true over <math>\mathbb{R}</math>.

The [[Jordan–Chevalley decomposition]] expresses an operator as the sum of its semisimple (i.e., diagonalizable) part and its [[nilpotent]] part. Hence, a matrix is diagonalizable if and only if its nilpotent part is zero. Put in another way, a matrix is diagonalizable if each block in its [[Jordan form]] has no nilpotent part; i.e., each "block" is a one-by-one matrix.

== Diagonalization ==
[[File:Diagonalization as rotation.gif|400px|thumb|right|The diagonalization of a matrix can be interpreted as a rotation of the axis to align them with the eigenvectors.]]
If a matrix <math>A</math> can be diagonalized, that is,

: <math>P^{-1}AP = \begin{pmatrix}
  \lambda_1 &         0 &  \dots &         0 \\
          0 & \lambda_2 &  \dots &         0 \\
     \vdots &    \vdots & \ddots &    \vdots \\
          0 &         0 &  \dots & \lambda_n
\end{pmatrix},</math>

then:

: <math>AP = P\begin{pmatrix}
  \lambda_1 &         0 &  \dots &         0 \\
          0 & \lambda_2 &  \dots &         0 \\
     \vdots &    \vdots & \ddots &    \vdots \\
          0 &         0 &  \dots & \lambda_n
\end{pmatrix}.</math>

Writing <math>P</math> as a [[block matrix]] of its column vectors <math>\vec{\alpha}_{i}</math>

:<math>P = \begin{pmatrix}\vec{\alpha}_1 & \vec{\alpha}_2 & \cdots & \vec{\alpha}_n\end{pmatrix},</math>

the above equation can be rewritten as

:<math>A\vec{\alpha}_i = \lambda_i\vec{\alpha}_i\qquad(i=1,2,\cdots,n).</math>

So the column vectors of <math>P</math> are [[right eigenvector]]s of <math>A</math>, and the corresponding diagonal entry is the corresponding [[eigenvalue]]. The invertibility of <math>P</math> also suggests that the eigenvectors are [[linearly independent]] and form a basis of <math>F^{n}</math>. This is the necessary and sufficient condition for diagonalizability and the canonical approach of diagonalization. The [[row vector]]s of <math>P^{-1}</math> are the [[left eigenvector]]s of <math>A</math>.

When a [[complex number|complex]] matrix <math>A</math><ref group=lower-alpha><math>A</math> in <math>\mathbb{C}^{n \times n}</math></ref> is a [[Hermitian matrix]] (or a [[real number|real]] matrix<ref group=lower-alpha><math>A</math> in <math>\mathbb{R}^{n \times n}</math></ref>, a [[symmetric matrix]]), eigenvectors of <math>A</math> can be chosen to form an [[orthonormal basis]] of <math>\mathbb{C}^n</math> (or <math>\mathbb{R}^n</math> for a real matrix). Under such circumstance <math>P</math> will be a [[unitary matrix]] (resp. [[orthogonal matrix]]) and <math>P^{-1}</math> equals the [[conjugate transpose]] of <math>P</math> (if real, then the [[transpose]] of <math>P</math>).

For most practical work matrices are diagonalized numerically using computer software. [[eigenvalue algorithm|Many algorithms]] exist to accomplish this.

== Simultaneous diagonalization ==
{{See also|Triangular matrix#Simultaneous triangularisability|l1=Simultaneous triangularisability|Weight (representation theory)|Positive definite matrix#Simultaneous_diagonalization|l3=Positive definite matrix}}

A set of matrices is said to be ''simultaneously diagonalizable'' if there exists a single invertible matrix <math>P</math> such that <math>P^{-1}AP</math> is a diagonal matrix for every <math>A</math> in the set. The following theorem characterises simultaneously diagonalizable matrices: A set of diagonalizable [[Commuting matrices|matrices commutes]] if and only if the set is simultaneously diagonalizable.<ref name="HornJohnson"/>{{rp|pp. 51-53}}

The set of all <math>n \times n</math> diagonalizable matrices (over <math>\mathbb{C}</math>) with <math>n > 1</math> is not simultaneously diagonalizable. For instance, the matrices

:<math> \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} </math>

are diagonalizable but not simultaneously diagonalizable because they do not commute.

A set consists of commuting [[normal matrix|normal matrices]] if and only if it is simultaneously diagonalizable by a [[unitary matrix]]; that is, there exists a unitary matrix <math>U</math> such that <math>U^{*}AU</math> is diagonal for every <math>A</math> in the set.

In the language of [[Lie theory]], a set of simultaneously diagonalizable matrices generate a [[toral Lie algebra]].

== Examples ==

=== Diagonalizable matrices ===
* [[Involution (mathematics)|Involution]]s are diagonalizable over the reals (and indeed any field of characteristic not 2), with ±1 on the diagonal.
* Finite order [[endomorphism]]s are diagonalisable over <math>\mathbb{C}</math> (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with [[roots of unity]] on the diagonal. This follows since the minimal polynomial is [[separable polynomial|separable]], because the roots of unity are distinct.
* [[Projection (linear algebra)|Projections]] are diagonalizable, with 0s and 1s on the diagonal.
* Real [[symmetric matrices]] are diagonalizable by [[orthogonal matrix|orthogonal matrices]]; i.e., given a real symmetric matrix <math>A</math>, <math>Q^{\mathrm T}AQ</math> is diagonal for some orthogonal matrix <math>Q</math>. More generally, matrices are diagonalizable by [[unitary matrix|unitary matrices]] if and only if they are [[normal matrix|normal]]. In the case of the real symmetric matrix, we see that <math>A=A^{\mathrm T}</math>, so clearly <math>AA^{\mathrm T} = A^{\mathrm T}A</math> holds. Examples of normal matrices are real symmetric (or [[Skew-symmetric matrix|skew-symmetric]]) matrices (e.g. covariance matrices) and [[Hermitian matrix|Hermitian matrices]] (or skew-Hermitian matrices). See [[spectral theorem]]s for generalizations to infinite-dimensional vector spaces.

=== Matrices that are not diagonalizable ===
In general, a [[rotation matrix]] is not diagonalizable over the reals, but all [[rotation matrix#Independent planes|rotation matrices]] are diagonalizable over the complex field. Even if a matrix is not diagonalizable, it is always possible to "do the best one can", and find a matrix with the same properties consisting of eigenvalues on the leading diagonal, and either ones or zeroes on the superdiagonal – known as [[Jordan Normal Form|Jordan normal form]].

Some matrices are not diagonalizable over any field, most notably nonzero [[nilpotent matrix|nilpotent matrices]]. This happens more generally if the [[Eigenvalue, eigenvector and eigenspace#Algebraic and geometric multiplicities|algebraic and geometric multiplicities]] of an eigenvalue do not coincide. For instance, consider

:<math> C = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. </math>

This matrix is not diagonalizable: there is no matrix <math>U</math> such that <math>U^{-1}CU</math> is a diagonal matrix. Indeed, <math>C</math> has one eigenvalue (namely zero) and this eigenvalue has algebraic multiplicity 2 and geometric multiplicity 1.

Some real matrices are not diagonalizable over the reals. Consider for instance the matrix

:<math> B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}. </math>

The matrix <math>B</math> does not have any real eigenvalues, so there is no '''real''' matrix <math>Q</math> such that <math>Q^{-1}BQ</math> is a diagonal matrix. However, we can diagonalize <math>B</math> if we allow complex numbers. Indeed, if we take

:<math> Q = \begin{bmatrix} 1 & \textrm{i} \\ \textrm{i} & 1 \end{bmatrix}, </math>

then <math>Q^{-1}BQ</math> is diagonal. It is easy to find that B is the rotation matrix which rotates counterclockwise by angle <math>\theta=\tfrac{3\pi}{2}</math>

Note that the above examples show that the sum of diagonalizable matrices need not be diagonalizable.

=== How to diagonalize a matrix ===
Consider a matrix

:<math>A=\begin{bmatrix}
 1& 2  & 0 \\
0 & 3  & 0 \\
2 & -4 & 2 \end{bmatrix}.</math>

This matrix has [[eigenvalues]]

:<math> \lambda_1 = 3, \quad \lambda_2 = 2, \quad \lambda_3= 1. </math>

<math>A</math> is a <math>3 \times 3</math> matrix with 3 different eigenvalues; therefore, it is diagonalizable. Note that if there are exactly ''n'' distinct eigenvalues in an <math>n \times n</math> matrix then this matrix is diagonalizable.

These eigenvalues are the values that will appear in the diagonalized form of matrix ''A'', so by finding the eigenvalues of ''A'' we have diagonalized it. We could stop here, but it is a good check to use the [[eigenvectors]] to diagonalize ''A''.

The [[eigenvectors]] of <math>A</math> are

:<math>v_1 = \begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix}.</math>

One can easily check that <math>A v_k = \lambda_k v_k.</math>

Now, let <math>P</math> be the matrix with these eigenvectors as its columns:

:<math>P= \begin{bmatrix}
-1 & 0 & -1 \\
-1 & 0  & 0 \\
2 & 1 & 2 \end{bmatrix}.</math>

Note that there is no preferred order of the eigenvectors in <math>P</math>; changing the order of the [[eigenvectors]] in <math>P</math> just changes the order of the [[eigenvalues]] in the diagonalized form of <math>A</math>.<ref>{{cite book| last1=Anton |first1=H. |last2= Rorres|first2= C. |title=Elementary Linear Algebra (Applications Version) | url=https://archive.org/details/studentsolutions00grob | url-access=registration |publisher=John Wiley & Sons|edition=8th|date=22 Feb 2000| ISBN= 978-0-471-17052-5}}</ref>

Then <math>P</math> diagonalizes <math>A</math>, as a simple computation confirms, having calculated <math>P^{-1}</math> using any [[Invertible matrix#Methods of matrix inversion|suitable method]]:

:<math>P^{-1}AP =
\begin{bmatrix}
0 & -1 & 0 \\
2 & 0  & 1 \\
-1 & 1 & 0 \end{bmatrix}
\begin{bmatrix}
1 & 2  & 0 \\
0 & 3  & 0 \\
2 & -4 & 2 \end{bmatrix}
\begin{bmatrix}
-1 & 0 & -1 \\
-1 & 0  & 0 \\
2 & 1 & 2 \end{bmatrix} =
\begin{bmatrix}
3 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 1\end{bmatrix}.</math>

In fact, this follows abstractly from the fact that for the standard basis <math>e_1, e_2, e_3</math> we have 
:<math> P^{-1} A P e_k = P^{-1} A v_k = P^{-1} \lambda_k v_k = \lambda_k e_k. </math>
where we make use of the fact that <math>P e_k = v_k</math> is the k-th column of <math>P</math> and hence <math>P^{-1}v_k = e_k</math>. Note that the eigenvalues <math>\lambda_k</math> appear in the diagonal matrix.

== An application ==
Diagonalization can be used to compute the powers of a matrix <math>A</math> efficiently, provided the matrix is diagonalizable. Suppose we have found that

:<math>P^{-1}AP = D \Rightarrow  PP^{-1}APP^{-1} = PDP^{-1} \Rightarrow A = PDP^{-1}</math>

where <math>D</math> is a diagonal matrix. Then, as the matrix product is associative,

:<math>\begin{align} 
  A^k &= \left(PDP^{-1}\right)^k = \left(PDP^{-1}\right) \cdot \left(PDP^{-1}\right) \cdots \left(PDP^{-1}\right) \\
      &= PD\left(P^{-1}P\right) D \left(P^{-1}P\right) \cdots \left(P^{-1}P\right) D P^{-1} \\
      &= PD^kP^{-1}
\end{align}</math>

and the latter is easy to calculate since it only involves the powers of a diagonal matrix. This approach can be generalized to [[matrix exponential]] and other [[matrix function]]s that can be defined as power series.

This is particularly useful in finding closed form expressions for terms of [[linear recursive sequences]], such as the [[Fibonacci number]]s{{Explain|date=January 2019}}.

=== Particular application ===
For example, consider the following matrix:

:<math>M = \begin{bmatrix}a & b - a\\ 0 & b\end{bmatrix}.</math>

Calculating the various powers of <math>M</math> reveals a  surprising pattern:

:<math>
  M^2 = \begin{bmatrix}a^2 & b^2-a^2 \\ 0 &b^2 \end{bmatrix},\quad
  M^3 = \begin{bmatrix}a^3 & b^3-a^3 \\ 0 &b^3 \end{bmatrix},\quad
  M^4 = \begin{bmatrix}a^4 & b^4-a^4 \\ 0 &b^4 \end{bmatrix},\quad
  \ldots
</math>

The above phenomenon can be explained by diagonalizing <math>M</math>.  To accomplish this, we need a basis of <math>\mathbb{R}^2</math> consisting of eigenvectors of <math>M</math>.  One such eigenvector basis is given by

:<math>
  \mathbf{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \mathbf{e}_1,\quad
  \mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \mathbf{e}_1 + \mathbf{e}_2,
</math>

where '''e'''<sub>''i''</sub> denotes the standard basis of '''R'''<sup>''n''</sup>. The reverse change of basis is given by

:<math>\mathbf{e}_1 = \mathbf{u},\qquad \mathbf{e}_2 = \mathbf{v} - \mathbf{u}.</math>

Straightforward calculations show that

:<math>M\mathbf{u} = a\mathbf{u},\qquad M\mathbf{v} = b\mathbf{v}.</math>

Thus, ''a'' and ''b'' are the eigenvalues corresponding to '''u''' and '''v''', respectively. By linearity of matrix multiplication, we have that

:<math> M^n \mathbf{u} = a^n\, \mathbf{u},\qquad M^n \mathbf{v} = b^n\,\mathbf{v}.</math>

Switching back to the standard basis, we have

:<math>\begin{align}
  M^n \mathbf{e}_1 &= M^n \mathbf{u} = a^n \mathbf{e}_1, \\
  M^n \mathbf{e}_2 &= M^n \left(\mathbf{v} - \mathbf{u}\right) = b^n \mathbf{v} - a^n\mathbf{u} = \left(b^n - a^n\right) \mathbf{e}_1 + b^n\mathbf{e}_2.
\end{align}</math>

The preceding relations, expressed in matrix form, are

:<math>M^n = \begin{bmatrix}a^n & b^n - a^n \\ 0 & b^n \end{bmatrix}, </math>

thereby explaining the above phenomenon.

== Quantum mechanical application ==
In [[quantum mechanics|quantum mechanical]] and [[quantum chemistry|quantum chemical]] computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent [[Schrödinger equation]] is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space (a [[Hilbert space]]).

A very common approximation is to truncate Hilbert space to finite dimension, after which the  Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian matrix. Formally this approximation is founded on the [[variational principle]], valid for Hamiltonians that are bounded from below.

[[Perturbation theory (quantum mechanics)#First order corrections|First-order perturbation theory]] also leads to matrix eigenvalue problem for degenerate states.

== See also ==
* [[Defective matrix]]
* [[Scaling (geometry)]]
* [[Triangular matrix]]
* [[Semisimple operator]]
* [[Diagonalizable group]]
* [[Jordan normal form]]
* [[Weight module]] – associative algebra generalization
* [[Orthogonal diagonalization]]

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== External links ==
* {{planetmath reference|id=1960|title=Diagonalization}} (or try https://planetmath.org/Diagonalization)

[[Category:Matrices]]