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Wikipedia / Diagonalizable matrix

In linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix such that . This is equivalent to (Such are not unique.) This property exists for any linear map: for a finite-dimensional vector space a linear map is called diagonalizable if there exists an ordered basis consisting of eigenvectors has a matrix form a basis consisting of eigenvectors of  are the corresponding eigenvalues of  is represented by

Diagonalization and  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

The geometric transformation represented by a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling). That is, it can 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 with real with real entries, but it is possible with complex 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 in the space of all matrices, which means any defective matrix can be deformed into a diagonalizable matrix by a small 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-simple matrices.

Definition

with entries in a field is called diagonalizable or nondefective invertible matrix (i.e. an element of the general linear group is a diagonal matrix.

Characterization

The fundamental fact about diagonalizable maps and matrices is expressed by the following:

  • is diagonalizable if and only if the sum of the dimensions, which is the case if and only if there exists a basis having these basis vectors is known as a modal matrix.

  • is diagonalizable if and only if the sum of the dimensionss of its eigenspaces is equal to will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of

The following sufficient (but not necessary) condition is often useful.

be a matrix over that is not an integer multiple of the characteristic of is annihilated by some polynomial which has no multiple root (since and is divided by the minimal polynomial of

matrices that are not diagonalizable over considered as a subset of has Lebesgue measure zero. One can also say that the diagonalizable matrices form a dense subset with respect to the Zariski topology: the non-diagonalizable matrices lie inside the vanishing set of the discriminant of the characteristic polynomial, which is a hypersurface. From that follows also density in the usual (strong) topology given by a norm. The same is not true over

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

which is written with respect to basis E. Suppose also that there exists the following eigen-equation:

, which changes basis from E to F as follows:

,

which changes basis from F to E instead and so we have the following relationship :

, which is still in the E-basis. Similarly, the diagonal matrix is in the F-basis.

can be diagonalized, that is,

then:

written in the basis of E so that we can represent P in block matrix form in the following manner:

In block matrix form, we can consider the A-matrix to be a matrix of 1x1 dimensions whilst P is a 1xn dimensional matrix. The D-matrix can be written in full form with all the diagonal elements as an nxn dimensional matrix:

Taking each component of the block matrix individually on both sides, we end up with the following:

are right eigenvectors of and the corresponding diagonal entry is the corresponding eigenvalue also suggests that the eigenvectors are linearly independent and form a basis of This is the necessary and sufficient condition for diagonalizability and the canonical approach of diagonalization. The row vectors are the left eigenvectors of

is a Hermitian matrix (or more generally a normal matrix can be chosen to form an orthonormal basis of can be chosen to be a unitary matrix is a real symmetric matrix can be chosen to be an orthogonal matrix.

For most practical work matrices are diagonalized numerically using computer software. Many algorithms exist to accomplish this.

Simultaneous diagonalization

A set of matrices is said to be simultaneously diagonalizable in the set. The following theorem characterizes simultaneously diagonalizable matrices: A set of diagonalizable matrices commutes if and only if the set is simultaneously diagonalizable.

diagonalizable matrices (over is not simultaneously diagonalizable. For instance, the matrices

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

A set consists of commuting normal matrices if and only if it is simultaneously diagonalizable by a unitary matrix in the set.

In the language of Lie theory, a set of simultaneously diagonalizable matrices generates a toral Lie algebra.

Examples

Diagonalizable matrices

  • Involutionss are diagonalizable over the reals (and indeed any field of characteristic not 2), with ±1 on the diagonal.

  • Finite order endomorphisms (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, because the roots of unity are distinct.

  • Projections are diagonalizable, with 0s and 1s on the diagonal.

  • Real symmetric matrices are diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix is diagonal for some orthogonal matrix More generally, matrices are diagonalizable by unitary matrices if and only if they are normal. In the case of the real symmetric matrix, we see that holds. Examples of normal matrices are real symmetric (or skew-symmetric) matrices (e.g. covariance matrices) and Hermitian matrices (or skew-Hermitian matrices). See spectral theorems for generalizations to infinite-dimensional vector spaces.

Additional geometric visualizations of orthogonal diagonalization, including reflection and orthogonal projection matrices, are available at Wikimedia Commons.

Matrices that are not diagonalizable

In general, a rotation matrix is not diagonalizable over the reals, but all 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.

Some matrices are not diagonalizable over any field, most notably nonzero nilpotent matrices. This happens more generally if the algebraic and geometric multiplicities of an eigenvalue do not coincide. For instance, consider

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

does not have any real eigenvalues, so there is no real if we allow complex numbers. Indeed, if we take

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

How to diagonalize a matrix

Diagonalizing a matrix is the same process as finding its eigenvalues and eigenvectors, in the case that the eigenvectors form a basis. For example, consider the matrix

The roots of the characteristic polynomial are the eigenvalues and gives for These vectors form a basis of so we can assemble them as the column vectors of a change-of-basis takes the standard basis to the eigenbasis, has the standard basis as its eigenvectors, which is the defining property of

Note that there is no preferred order of the eigenvectors in changing the order of the eigenvectors just changes the order of the eigenvalues in the diagonalized form of

Application to matrix functions

Diagonalization can be used to efficiently compute the powers of a matrix

in the example above we compute:

This approach can be generalized to matrix exponential and other matrix functions that can be defined as power series. For example, defining we have:

This is particularly useful in finding closed form expressions for terms of linear recursive sequences, such as the Fibonacci numbers.

Particular application

For example, consider the following matrix:

reveals a surprising pattern:

The above phenomenon can be explained by diagonalizing consisting of eigenvectors of 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 Hilbert space.

A very common approximation is to truncate (or project) the 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.

First-order perturbation theory also leads to matrix eigenvalue problem for degenerate states.

Operator theory

Matrices can be generalized to linear operators. A diagonal matrix can be generalized to diagonal operators on Hilbert spaces.

.

, define the p-Schatten norm is the trace. The p-Schatten class is the set of all operators with finite p-Schatten norm.

Weyl, von Neumann, and Kuroda, showed the following:.In other words, any self-adjoint operator is an infinitesimal perturbation from a diagonal operator, where "infinitesimal" is in the sense of p-Schatten norm. In particular, since the Hilbert–Schmidt operator class is the 2-Schatten class, this means that any self-adjoint operator is diagonalizable after a perturbation by an infinitesimal Hilbert–Schmidt operator. In fact, the above result could be further generalized:For any norm ideal.

(the trace class). This is a simple corollary of the Kato–Rosenblum have the same absolutely continuous part of the spectrum has no absolutely continuous part, then it can be diagonalized after perturbation by an infinitesimal trace class operator.

For simultaneous diagonalization

See also

References