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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 −1AP 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 a 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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
For the square matrix section, see square matrix. ...
Several equivalence relations in mathematics are called similarity. ...
In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...
In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
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. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
The Jordan-Chevalley decomposition expresses an operator as the sum of its diagonal part and its nilpotent part. In mathematics, the Jordan–Chevalley decomposition expresses a linear operator as the sum of its semisimple part and its nilpotent parts. ...
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...
Characterization
The fundamental fact about diagonalizable maps and matrices is expressed by the following: - An n-by-n matrix A over the field F is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n, which is the case if and only if there exists a basis of Fn consisting of eigenvectors of A. If such a basis has been found, one can form the matrix P having these basis vectors as columns, and P -1AP will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of A.
- A linear map T : V → V is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to dim(V), which is the case if and only if there exists a basis of V consisting of eigenvectors of T. With respect to such a basis, T will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of T.
Another characterization: A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0. ...
The following sufficient (but not necessary) condition is often useful.
- An n-by-n matrix A is diagonalizable over the field F if it has n distinct eigenvalues in F, i.e. if its characteristic polynomial has n distinct roots in F.
- A linear map T : V → V with n=dim(V) is diagonalizable if it has n distinct eigenvalues, i.e. if its characteristic polynomial has n distinct roots in F.
As a rule of thumb, over C almost every matrix is diagonalizable. More precisely: the set of complex n-by-n matrices that are not diagonalizable over C, considered as a subset of Cn×n, 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. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...
“Superset†redirects here. ...
In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around...
In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...
In mathematics, a hypersurface is some kind of submanifold. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
The same is not true over R. As n increases, it becomes (in some sense) less and less likely that a randomly selected real matrix is diagonalizable over R.
Examples
Diagonalizable matrices - Involutions are diagonalizable over the reals (and indeed any field of characteristic not 2), with 1's and -1's on the diagonal
- Finite order endomorphisms (including involutions) are diagonalizable over the complex numbers, or any algebraically closed field where the characteristic of the field doesn't divide the order of the endomorphism (as the roots of unity are distinct), with roots of unity on the diagonal. This is a part of representation theory of cyclic groups.
- Projections are diagonalizable, with 0's and 1's on the diagonal.
In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...
In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
The transformation P is the orthogonal projection onto the line m. ...
Matrices that are not diagonalizable Some matrices are not diagonalizable over any field, most notably nilpotent matrices. This happens more generally if the geometric and algebraic multiplicities of an eigenvalue do not coincide. For instance, consider In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...
Fig. ...
In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
 This matrix is not diagonalizable: there is no matrix U such that U − 1CU is a diagonal matrix. Indeed, C 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
 The matrix B does not have any real eigenvalues, so there is no real matrix Q such that Q − 1BQ is a diagonal matrix. However, we can diagonalize B if we allow complex numbers. Indeed, if we take  then Q − 1BQ is diagonal.
How to diagonalize a matrix Consider a matrix  This matrix has eigenvalues In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...
 So A is a 3-by-3 matrix with 3 different eigenvalues, therefore it is diagonalizable.
If we want to diagonalize A, we need to compute the corresponding eigenvectors. They are In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
 One can easily check that Avk = λkvk.
Now, let P be the matrix with these eigenvectors as its columns:
 Then P diagonalizes A, as a simple computation confirms:  Note that the eigenvalues λk appear in the diagonal matrix.
An application Diagonalization can be used to compute the powers of a matrix A efficiently, provided the matrix is diagonalizable. Suppose we have found that  is a diagonal matrix. Then, as the matrix product is associative,  and the latter is easy to calculate since it only involves the powers of a diagonal matrix.
This is particularly useful in finding closed form expressions for terms of a linear recursive sequences, such as the Fibonacci numbers. A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is...
Particular application For example, consider the following matrix:  Calculating the various powers of M reveals a surprising pattern:  The above phenomenon can be explained by diagonalizing M. To accomplish this, we need a basis of R2 consisting of eigenvectors of M. One such eigenvector basis is given by  where ei denotes the standard basis of Rn. The reverse change of basis is given by  Straightforward calculations show that  Thus, a and b are the eigenvalues corresponding to u and v, respectively. By linearity of matrix multiplication, we have that  Switching back to the standard basis, we have   The preceding relations, expressed in matrix form, are  thereby explaining the above phenomenon.
Quantum mechanical application In quantum mechanical and 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. But also first-order perturbation theory for degenerate states leads to a matrix eigenvalue problem. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...
Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ...
For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...
The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...
See also In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard...
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ...
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
External links PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
References - Roger A. Horn and Charles R. Johnson, Matrix Analysis, Chapter 1, Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
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