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  and a diagonal matrix such that. This is equivalent to 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 of  consisting of eigenvectors of. These definitions are equivalent: if  has a matrix representation as above, then the column vectors of  form a basis consisting of eigenvectors of and the diagonal entries of  are the corresponding eigenvalues of with respect to this eigenvector basis,  is represented by
Diagonalization is the process of finding the above  and and makes many subsequent computations easier. One can raise a diagonal matrix  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. 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. It can happen that a matrix with real entries is defective over the real numbers, meaning that is impossible for any invertible and diagonal with real entries, but it is possible with complex entries, so that 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. 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

A square matrix with entries in a field is called diagonalizable or nondefective if there exists an invertible matrix,, such that is a diagonal matrix.

Characterization

The fundamental fact about diagonalizable maps and matrices is expressed by the following:
  • An matrix over a field is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to, which is the case if and only if there exists a basis of consisting of eigenvectors of. If such a basis has been found, one can form the matrix having these basis vectors as columns, and will be a diagonal matrix whose diagonal entries are the eigenvalues of. The matrix is known as a modal matrix for.
  • A linear map is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to which is the case if and only if there exists a basis of consisting of eigenvectors of. With respect to such a basis, will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of
The following sufficient condition is often useful.
  • An matrix is diagonalizable over the field if it has distinct eigenvalues in i.e. if its characteristic polynomial has distinct roots in however, the converse may be false. Consider which has eigenvalues 1, 2, 2 and is diagonalizable with diagonal form topology given by a norm. The same is not true over
The Jordan–Chevalley decomposition expresses an operator as the sum of its semisimple 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

Consider the two following arbitrary bases and. Suppose that there exists a linear transformation represented by a matrix which is written with respect to basis E. Suppose also that there exists the following eigen-equation:
The alpha eigenvectors are written also with respect to the E basis. Since the set F is both a set of eigenvectors for matrix A and it spans some arbitrary vector space, then we say that there exists a matrix which is a diagonal matrix that is similar to. In other words, is a diagonalizable matrix if the matrix is written in the basis F. We perform the change of basis calculation using the transition matrix, which changes basis from E to F as follows:
where is the transition matrix from E-basis to F-basis. The inverse can then be equated to a new transition matrix which changes basis from F to E instead and so we have the following relationship :
Both and transition matrices are invertible. Thus we can manipulate the matrices in the following fashion:The matrix will be denoted as, which is still in the E-basis. Similarly, the diagonal matrix is in the F-basis.
If a matrix can be diagonalized, that is,
then:
The transition matrix S has the E-basis vectors as columns written in the basis F. Inversely, the inverse transition matrix P has F-basis vectors written in the basis of E so that we can represent P in block matrix form in the following manner:
as a result we can write:
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:
Performing the above matrix multiplication we end up with the following result:Taking each component of the block matrix individually on both sides, we end up with the following:
So the column vectors of are right eigenvectors of and the corresponding diagonal entry is the corresponding eigenvalue. The invertibility of 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 of are the left eigenvectors of
When a complex matrix is a Hermitian matrix, eigenvectors of can be chosen to form an orthonormal basis of and can be chosen to be a unitary matrix. If in addition, is a real symmetric matrix, then its eigenvectors can be chosen to be an orthonormal basis of and 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 if there exists a single invertible matrix such that is a diagonal matrix for every 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.
The set of all diagonalizable matrices (over with 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; that is, there exists a unitary matrix such that is diagonal for every in the set.
In the language of Lie theory, a set of simultaneously diagonalizable matrices generates a toral Lie algebra.

Examples

Diagonalizable matrices

  • Involutions are diagonalizable over the reals, with ±1 on the diagonal.
  • Finite order endomorphisms are diagonalizable over 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 so clearly holds. Examples of normal matrices are real symmetric matrices and Hermitian matrices. See spectral theorems 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 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
This matrix is not diagonalizable: there is no matrix such that is a diagonal matrix. Indeed, has one eigenvalue 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 does not have any real eigenvalues, so there is no real matrix such that is a diagonal matrix. However, we can diagonalize if we allow complex numbers. Indeed, if we take
then is diagonal. It is easy to find that is the rotation matrix which rotates counterclockwise by angle
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 Solving the linear system gives the eigenvectors and while gives that is, for These vectors form a basis of so we can assemble them as the column vectors of a change-of-basis matrix to get:
We may see this equation in terms of transformations: takes the standard basis to the eigenbasis, so we have:
so that 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 in just changes the order of the eigenvalues in the diagonalized form of