Diagonalizable matrix

In linear algebra, a square matrix formula_1 is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix formula_2 and a diagonal matrix formula_3 such that or equivalently (Such formula_3 are not unique.) For a finite-dimensional vector space a linear map formula_5 is called diagonalizable if there exists an ordered basis of formula_6 consisting of eigenvectors of formula_7. These definitions are equivalent: if formula_7 has a matrix representation formula_9 as above, then the column vectors of formula_2 form a basis consisting of eigenvectors of and the diagonal entries of formula_3 are the corresponding eigenvalues of with respect to this eigenvector basis, formula_1 is represented by Diagonalization is the process of finding the above formula_2 and 
Diagonalizable matrices and maps are especially easy for computations, once their eigenvalues and eigenvectors are known. One can raise a diagonal matrix formula_3 to a power by simply raising the diagonal entries to that power, and the determinant of a diagonal matrix is simply the product of all diagonal entries; such computations generalize easily to Geometrically, a diagonalizable matrix is an inhomogeneous dilation (or "anisotropic scaling") — it scales the space, as does a "homogeneous dilation", but by a different factor along each eigenvector axis, the factor given by the corresponding eigenvalue.
A square matrix that is not diagonalizable is called "defective". It can happen that a matrix formula_1 with real entries is defective over the real numbers, meaning that formula_9 is impossible for any invertible formula_2 and diagonal formula_3 with real entries, but it is possible with complex entries, so that formula_1 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 normal form theorem 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 formula_20 matrix formula_1 over a field formula_22 is called diagonalizable or nondefective if there exists an invertible matrix formula_2 such that formula_24 is a diagonal matrix. Formally,
Characterization.
The fundamental fact about diagonalizable maps and matrices is expressed by the following:
Another characterization: A matrix or linear map is diagonalizable over the field formula_22 if and only if its minimal polynomial is a product of distinct linear factors over (Put another way, a matrix is diagonalizable if and only if all of its elementary divisors are linear.)
The following sufficient (but not necessary) condition is often useful.
Let formula_1 be a matrix over If formula_1 is diagonalizable, then so is any power of it. Conversely, if formula_1 is invertible, formula_22 is algebraically closed, and formula_61 is diagonalizable for some formula_28 that is not an integer multiple of the characteristic of then formula_1 is diagonalizable. Proof: If formula_61 is diagonalizable, then formula_1 is annihilated by some polynomial which has no multiple root (since and is divided by the minimal polynomial of 
Over the complex numbers formula_66, almost every matrix is diagonalizable. More precisely: the set of complex formula_20 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.
If a matrix formula_1 can be diagonalized, that is,
then:
Writing formula_2 as a block matrix of its column vectors formula_72
the above equation can be rewritten as
So the column vectors of formula_2 are right eigenvectors of and the corresponding diagonal entry is the corresponding eigenvalue. The invertibility of formula_2 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 formula_77 are the left eigenvectors of 
When a complex matrix formula_78 is a Hermitian matrix (or more generally a normal matrix), eigenvectors of formula_1 can be chosen to form an orthonormal basis of and formula_2 can be chosen to be a unitary matrix. If in addition, formula_81 is a real symmetric matrix, then its eigenvectors can be chosen to be an orthonormal basis of formula_82 and formula_2 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 formula_2 such that formula_24 is a diagonal matrix for every formula_1 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 formula_20 diagonalizable matrices (over with formula_88 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 formula_90 such that formula_91 is diagonal for every formula_1 in the set.
In the language of Lie theory, a set of simultaneously diagonalizable matrices generate a toral Lie algebra.
Examples.
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 formula_90 such that formula_98 is a diagonal matrix. Indeed, formula_99 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 formula_101 does not have any real eigenvalues, so there is no real matrix formula_102 such that formula_103 is a diagonal matrix. However, we can diagonalize formula_101 if we allow complex numbers. Indeed, if we take
then formula_103 is diagonal. It is easy to find that formula_101 is the rotation matrix which rotates counterclockwise by angle formula_108
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 formula_110 are the eigenvalues Solving the linear system formula_111 gives the eigenvectors formula_112 and while formula_113 gives that is, formula_114 for These vectors form a basis of so we can assemble them as the column vectors of a change-of-basis matrix formula_2 to get:
formula_116
We may see this equation in terms of transformations: formula_2 takes the standard basis to the eigenbasis, so we have:
formula_118
so that formula_119 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 formula_2 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 
and the latter is easy to calculate since it only involves the powers of a diagonal matrix. For example, for the matrix formula_1 with eigenvalues formula_123 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:
Calculating the various powers of formula_127 reveals a surprising pattern:
The above phenomenon can be explained by diagonalizing To accomplish this, we need a basis of formula_129 consisting of eigenvectors of One such eigenvector basis is given by
where e"i" denotes the standard basis of R"n". 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.
First-order perturbation theory also leads to matrix eigenvalue problem for degenerate states.