The diagonalizable operator is the special case of this in which Af for each i.

By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

) Now Af is a diagonalizable operator which is also nilpotent.

but then since the operator is diagonalizable, the minimal polynomial cannot have a repeated root ; ;

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

A compact normal operator ( in particular, a normal operator on a finite-dimensional linear space ) is unitarily diagonalizable.

This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator.

The Jordan – Chevalley decomposition expresses an operator as the sum of its semisimple ( i. e., diagonalizable ) part and its nilpotent part.

If the operator is orthogonal ( an orthogonal involution ), it is orthonormally diagonalizable.

We have just observed that we can write Af where D is diagonalizable and N is nilpotent, and where D and N not only commute but are polynomials in T.

Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.

* For any abelian group A, one can form the corresponding diagonalizable group D ( A ), defined as a functor by setting D ( A )( T ) to be the set of abelian group homomorphisms from A to invertible global sections of O < sub > T </ sub > for each S-scheme T. If S is affine, D ( A ) can be formed as the spectrum of a group ring.

It can be described as the diagonalizable group D ( Z ) associated to the integers.

If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D ( G ), and vice versa.

Now suppose that we also have Af where D' is diagonalizable, N' is nilpotent, and Af.

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., one-by-one matrix.

Some matrices are not diagonalizable over any field, most notably nonzero nilpotent matrices.

* The only nilpotent diagonalizable matrix is the zero matrix.

The matrix P is Hermitian, therefore diagonalizable, so it is the identity matrix — in other words the columns of M are an orthonormal set and the columns of N are an orthogonal set.

It has long been known that if the space of illuminants can be described as a linear model with N basis terms, the proper color transformation will be the weighted sum of N fixed linear transformations, not necessarily consistently diagonalizable.

Furthermore the two are simultaneously diagonalizable, that is: both A and B are made diagonal by the same unitary matrix U. Both UAU < sup >*</ sup > and UBU < sup >*</ sup > are diagonal matrices.

This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.

Not all matrices are diagonalizable, but for matrices with complex coefficients many of them are: the set of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices ( for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have multiple roots ).

Such matrices are said to be diagonalizable.

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. ( Put in another way, a matrix is diagonalizable if and only if all of its elementary divisors are linear.

More precisely: the set of complex n-by-n matrices that are not diagonalizable over C, considered as a subset of C < sup > n × n </ sup >, has Lebesgue measure zero.

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

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

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 in each direction, determined by the scale factors on each axis ( diagonal entries ).

# A is diagonalizable by a unitary matrix.

While this provides a valid proof ( for matrices over the complex numbers ), the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix ( diagonalizable or not ), nor on the kind of entries allowed ( for matrices with real entries the diagonizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley – Hamilton theorem holds for them ).

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 T which is represented by a diagonal matrix.

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

* 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.

Proof: If is diagonalizable, then A is annihilated by some polynomial, which has no multiple root ( since ) and is divided by the minimal polynomial of A.

* 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.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

For the eigenvalue problem, Bauer and Fike proved that if is an eigenvalue for a diagonalizable matrix with eigenvector matrix, then the absolute error in calculating is bounded by the product of and the absolute error in.

The result was proved by using measure classification theorem for diagonalizable actions of higher-rank groups, and an isolation theorem proved by Lindenstrauss and Barak Weiss.

The 2 × 2 matrix is real symmetric and so ( by the spectral theorem ) it is " orthogonally diagonalizable ".

A matrix is diagonalizable by a suitable choice of coordinates if and only if there is an eigenbasis ; if a matrix is not diagonalizable, it is said to be defective.