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Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their Eigenvalue s and Eigenvector s are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power.

The Jordan-Chevalley Decomposition expresses an operator as the sum of its diagonal part and its Nilpotent part.


CHARACTERIZATION

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 Dimension s of its eigenspaces is equal to ''n'', which is the case if and only if there exists a Basis of ''F''''n'' 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'' -1''AP'' 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 Dimension s 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''.

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 of the characteristic polynomial vanishes, which is a Hypersurface . From that follows also density in the usual (''strong'') topology given by a Norm .

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



Diagonalizable matrices

  • Involution s are diagonalizable over the reals (and indeed any field of characteristic not 2), with 1's and -1's on the diagonal

  • Finite order endomorphisms (including involutions) are diagonalizable over the complex numbers, or any algebraically closed field where the characteristic of the field doesn't divide the order of the endomorphism (as the roots of unity are distinct), with Roots Of Unity on the diagonal. This is a part of Representation Theory of cyclic groups.

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



Matrices that are not diagonalizable


Some matrices are not diagonalizable over any field, most notably Nilpotent matrices.
This happens more generally if the Geometric and Algebraic Multiplicities of an eigenvalue do not coincide. For instance, consider
: C = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}.
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.

Some real matrices are not diagonalizable over the reals. Consider for instance the matrix
: B = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}.
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
: Q = \begin{bmatrix} 1 & extrm{i} \ extrm{i} & 1 \end{bmatrix},
then Q^{-1}BQ is diagonal.


How to diagonalize a matrix


Consider a matrix
:A=\begin{bmatrix}
1 & 2 & 0 \
0 & 3 & 0 \
2 & -4 & 2 \end{bmatrix}.

This matrix has Eigenvalues
: \lambda_1 = 3, \quad \lambda_2 = 2, \quad \lambda_3= 1.
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
: v_1 = \begin{bmatrix} -1 \ -1 \ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -1 \ 0 \ 2 \end{bmatrix}.
One can easily check that A v_k = \lambda_k v_k.

Now, let ''P'' be the matrix with these eigenvectors as its columns:
:P=
\begin{bmatrix}
-1 & 0 & -1 \
-1 & 0 & 0 \
2 & 1 & 2 \end{bmatrix}.
Then ''P'' diagonalizes ''A'', as a simple computation confirms:
:P^{-1}AP =
\begin{bmatrix}
0 & -1 & 0 \
2 & 0 & 1 \
-1 & 1 & 0 \end{bmatrix}
\begin{bmatrix}
1 & 2 & 0 \
0 & 3 & 0 \
2 & -4 & 2 \end{bmatrix}
\begin{bmatrix}
-1 & 0 & -1 \
-1 & 0 & 0 \
2 & 1 & 2 \end{bmatrix} =
\begin{bmatrix}
3 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 1\end{bmatrix}.
Note that the eigenvalues \lambda_k appear in the diagonal matrix.


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, as the matrix product is associative,

:\begin{align} A^k &= (PDP^{-1})^k = (PDP^{-1}) \cdot (PDP^{-1}) \cdots (PDP^{-1}) \
&= PD(P^{-1}P) D (P^{-1}P) \cdots (P^{-1}P) D P^{-1} = PD^kP^{-1} \end{align}

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

This is particularly useful in finding closed form expressions for terms of a Linear Recursive Sequences , such as the Fibonacci Number s.


Particular application

For example, consider the following matrix:
:M =\begin{bmatrix}a & b-a \ 0 &b \end{bmatrix}.
Calculating the various powers of ''M'' reveals a surprising pattern:
:
M^2 = \begin{bmatrix}a^2 & b^2-a^2 \ 0 &b^2 \end{bmatrix},\quad
M^3 = \begin{bmatrix}a^3 & b^3-a^3 \ 0 &b^3 \end{bmatrix},\quad
M^4 = \begin{bmatrix}a^4 & b^4-a^4 \ 0 &b^4 \end{bmatrix},\quad \ldots


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
:\mathbf{u}=\begin{bmatrix} 1 \ 0 \end{bmatrix}=\mathbf{e}_1,\quad
\mathbf{v}=\begin{bmatrix} 1 \ 1 \end{bmatrix}=\mathbf{e}_1+\mathbf{e}_2,
where ei denotes the standard basis of '''R'''n.
The reverse change of basis is given by
: \mathbf{e}_1 = \mathbf{u},\qquad \mathbf{e}_2 = \mathbf{v}-\mathbf{u}.

Straightforward calculations show that
:M\mathbf{u} = a\mathbf{u},\qquad M\mathbf{v}=b\mathbf{v}.
Thus, ''a'' and ''b'' are the eigenvalues corresponding to u and '''v''', respectively.
By linearity of matrix multiplication, we have that
: M^n \mathbf{u} = a^n\, \mathbf{u},\qquad M^n \mathbf{v}=b^n\,\mathbf{v}.
Switching back to the standard basis, we have
: M^n \mathbf{e}_1 = M^n \mathbf{u} = a^n \mathbf{e}_1,
: M^n \mathbf{e}_2 = M^n (\mathbf{v}-\mathbf{u}) = b^n \mathbf{v} - a^n\mathbf{u} = (b^n-a^n) \mathbf{e}_1+b^n\mathbf{e}_2.
The preceding relations, expressed in matrix form, are
:
M^n = \begin{bmatrix}a^n & b^n-a^n \ 0 &b^n \end{bmatrix},

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.
But also first-order perturbation theory for degenerate states leads to a matrix eigenvalue problem.


SEE ALSO



EXTERNAL LINKS




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).