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In mathematics, a [[complex number|complex]] [[Matrix_(mathematics)#Square_matrices|square]] [[matrix (mathematics)|matrix]] ''A'' is '''normal''' if

:<math>A^*A=AA^* \ </math>

where ''A''* is the [[conjugate transpose]] of ''A''. That is, a matrix is normal if it [[Commute (mathematics)|commutes]] with its conjugate transpose.

A matrix ''A'' with [[real number|real]] entries satisfies ''A''*=''A''T, and is therefore normal if ''A''T''A'' = ''AA''T.

Normality is a convenient test for [[diagonalizable|diagonalizability]]: a matrix is normal if and only if it is [[Similar matrix|unitarily similar]] to a [[diagonal matrix]], and therefore any matrix ''A'' satisfying the equation ''A''*''A''=''AA''* is diagonalizable.

The concept of normal matrices can be extended to [[normal operator]]s on infinite dimensional [[Hilbert space]]s and to normal elements in [[C*-algebra]]s. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

==Special cases==

Among complex matrices, all [[unitary matrix|unitary]], [[hermitian matrix|Hermitian]], and [[Skew-Hermitian matrix|skew-Hermitian]] matrices are normal. Likewise, among real matrices, all [[Orthogonal matrix|orthogonal]], [[Symmetric matrix|symmetric]], and [[skew-symmetric matrix|skew-symmetric]] matrices are normal.

However, it is ''not'' the case that all normal matrices are either unitary or (skew-)Hermitian. As an example, the matrix

:<math>A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}</math>

is normal because

:<math>AA^* = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix} = A^*A.</math>

The matrix ''A'' is neither unitary, Hermitian, nor skew-Hermitian.


The sum or product of two normal matrices is not necessarily normal. If they commute, however, then this is true.

If ''A'' is both a [[triangular matrix]] and a normal matrix, then ''A'' is [[diagonal matrix|diagonal]]. This can be seen by looking at the diagonal entries of ''A''*''A'' and ''AA''*, where ''A'' is a normal, triangular matrix. Say ''A'' is upper-triangular. Because ''(A''*''A)ii''=''(AA''*'')ii'', the first row must have the same norm as the first column,
:<math>||A e_1||^2 = ||A^* e_1||^2</math>.
The first entry of row 1 and column 1 are the same, and the column 1 is zero for entries 2 through ''n''. This implies the first row must be zero for entries 2 through n. Continuing this argument for row column pairs 2 through n shows ''A'' is diagonal.

== Consequences ==
The concept of normality is important because normal matrices are precisely those to which the [[spectral theorem]] applies: a matrix ''A'' is normal if and only if it can be represented by a [[diagonal matrix]] Λ and a [[unitary matrix]] ''U'' by the formula

:<math> \mathbf{A} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^* </math>
where
:<math> \mathbf{\Lambda} = \operatorname{diag}(\lambda_1, \lambda_2, \dots)</math>
:<math> \mathbf{U}^*\mathbf{U} = \mathbf{U} \mathbf{U}^* = \mathbf{I}.</math>

The entries λ of the diagonal matrix Λ are the [[eigenvalue]]s of ''A'', and the columns of ''U'' are the [[eigenvector]]s of ''A''. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of ''U''.

Another way of stating the [[spectral theorem]] is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen [[orthonormal basis]] of '''C'''''n''. Phrased differently: a matrix is normal if and only if its [[eigenspace]]s span '''C'''''n'' and are pairwise [[orthogonal]] with respect to the standard inner product of '''C'''''n''.

The spectral theorem for normal matrices can be seen as a special case of the more general result which holds for all square matrices: [[Schur decomposition]]. In fact, let ''A'' be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If ''A'' is normal, so is ''B''. But then ''B'' must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal.

The spectral theorem permits the classification of normal matrices in terms of their spectra. For example, a normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane. Also, a normal matrix is [[self-adjoint]] if and only if its spectrum consists of reals.

In general, the sum or product of two normal matrices need not be normal. However, there is a special case: if ''A'' and ''B'' are normal with ''AB'' = ''BA'', then both ''AB'' and ''A'' + ''B'' are also normal. Furthermore the two are ''[[simultaneously diagonalizable]]'', that is: both ''A'' and ''B'' are made diagonal by the same unitary matrix ''U''. Both ''UAU*'' and ''UBU*'' are diagonal matrices. In this special case, the columns of ''U*'' are eigenvectors of both ''A'' and ''B'' and form an orthonormal basis in '''C'''''n''. 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.

== Equivalent definitions ==
It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let ''A'' be a ''n''-by-''n'' complex matrix. Then the following are equivalent:
# ''A'' is normal.
# ''A'' is [[diagonalizable matrix|diagonalizable]] by a unitary matrix.
# The entire space is spanned by some orthonormal set of eigenvectors of ''A''. {{citation needed|date=January 2013}}
# <math>\|Ax\| = \|A^*x\|</math> for every ''x''.
# <math>\operatorname{tr} (A^* A) = \sum_j^n |\lambda_j|^2.</math> (That is, the [[Frobenius norm]] of ''A'' can be computed by the eigenvalues of ''A''.)
# The [[Hermitian matrix|Hermitian]] part <math>\frac{1}{2} (A + A^*)</math> and [[Skew-Hermitian matrix|skew-Hermitian]] part <math>\frac{1}{2} (A - A^*)</math> of ''A'' commute.
# <math>A^*</math> is a polynomial (of degree ≤ n − 1) in <math>A</math>.<ref>Proof: When ''A'' is normal, use [[Lagrange polynomial|Lagrange's interpolation]] formula to construct a polynomial ''P'' such that <math>\overline{\lambda_j} = P(\lambda_j)</math>, where <math>\lambda_j</math> are the eigenvalues of ''A''.</ref>
# <math>A^* = AU</math> for some unitary matrix ''U''.<ref>Horn, pp. 109</ref>
# ''U'' and ''P'' commute, where we have the [[polar decomposition]] ''A'' = ''UP'' with a unitary matrix ''U'' and some [[positive-definite matrix|positive semidefinite matrix]] ''P''.
# ''A'' commutes with some normal matrix ''N'' with distinct eigenvalues.
#<math>\sigma_i(A)=|\lambda_i(A)|</math> for all <math>i=1...n</math> where ''A'' has [[singular values]] <math>\sigma_1(A)\ge ...\ge\sigma_n(A)</math> and eigenvalues <math>|\lambda_1(A)|\ge ...\ge|\lambda_n(A)|</math><ref name=book>{{Cite book|title=Topics in Matrix Analysis|first1=Roger A.|last1=Horn|first2=Charles R.|last2=Johnson|publisher=Cambridge University Press|year=1991|isbn=978-0-521-30587-7|page=157}}</ref>

Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is only [[quasinormal operator|quasinormal]].

The operator norm of a normal matrix ''N'' equals the [[numerical radius|numerical]] and [[spectral radius|spectral radii]] of ''N''. (This fact generalizes to [[normal operator]]s.) Explicitly, this means:
:<math>\sup_{ \|x\|=1 } \|Nx\| = \sup_{ \|x\|=1 } |\langle Nx, x \rangle| = \max \{ |\lambda| : \lambda \in \sigma(N) \}</math>

==Analogy==
It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:
* [[Inverse matrix|Invertible matrices]] are analogous to non-zero [[complex number]]s
* The [[conjugate transpose]] is analogous to the [[complex conjugate]]
* [[Unitary matrix|Unitary matrices]] are analogous to [[complex number]]s whose [[absolute value]] is 1
* [[Hermitian matrix|Hermitian matrices]] are analogous to [[real number]]s
* Hermitian [[Positive-definite matrix|positive definite matrices]] are analogous to positive real numbers
* [[Skew-Hermitian matrix|Skew Hermitian matrices]] are analogous to purely [[imaginary number]]s

(As a special case, the complex numbers may be embedded in the normal <math>2\times 2</math> real matrices by the mapping <math>a+bi \mapsto \begin{pmatrix}a&b\\-b&a\end{pmatrix}</math>, which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.)

== Notes ==
{{reflist}}

== References ==
* {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-38632-6 | year=1985}}.

[[Category:Matrices]]

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Stepper motor - Revision history

en>Wtshymanski: out of place and to specific; wp:prose Undid revision 640953465 by Gayathri sowmya (talk)

180.149.51.235 at 06:54, 25 February 2014

en>Materialscientist: Reverted 1 edit by 49.147.71.5 identified as test/vandalism using STiki