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 (or linear spaces), 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 only the entries in the main diagonal are non-zero. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, 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. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... 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 mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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 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 mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, the dimension of a vector space V is the cardinality (i. ... The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 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 C^n×n, is a null set with respect to the Lebesgue measure. 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. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In measure theory, a null set is a set that it is negligible for the purposes of the measure in question. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the x-axis). ... In mathematics, a hypersurface is some kind of submanifold. ... The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ...

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

How to diagonalize a matrix

Consider a matrix

This matrix has eigenvalues

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

One can easily check that Av_k = λ_kv_k.

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 λ_k appear in the diagonal matrix.

Matrices that are not diagonalizable

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 ^{− 1}BQ is a diagonal matrix. However, we can diagonalize B if we allow complex numbers. Indeed, if we take

then Q ^{− 1}BQ is diagonal.

However, there are also matrices that are not diagonalizable, even if complex numbers are used. This happens if the geometric and algebraic multiplicities of an eigenvalues do not coincide. For instance, consider

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

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

P ^{− 1}AP = D

is a diagonal matrix. Then

A^k = (PDP ^{− 1})^k = PD^kP ^{− 1}

and the latter is easy to calculate since it only involves the powers of a diagonal matrix.

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 e_i 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.

See also

• Jordan form
• Scaling (geometry)

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, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. ...

External links

• Diagonalization on PlanetMath.

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).