Normal matrix
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In mathematics, a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
is normal if it
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
s with its
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
: The concept of normal matrices can be extended to
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal opera ...
s on infinite dimensional normed spaces and to normal elements in C*-algebras. 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. The
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
states that a matrix is normal if and only if it is unitarily similar to a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
, and therefore any matrix satisfying the equation is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces. The left and right singular vectors in the
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
of a normal matrix \mathbf = \mathbf \boldsymbol \mathbf^* differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.


Special cases

Among complex matrices, all
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
,
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
, and
skew-Hermitian __NOTOC__ In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relatio ...
matrices are normal, with all eigenvalues being unit modulus, real, and imaginary, respectively. Likewise, among real matrices, all orthogonal,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and skew-symmetric matrices are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it is ''not'' the case that all normal matrices are either unitary or (skew-)Hermitian, as their eigenvalues can be any complex number, in general. For example, A = \begin 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end is neither unitary, Hermitian, nor skew-Hermitian, because its eigenvalues are 2, (1\pm i\sqrt)/2; yet it is normal because AA^* = \begin 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end = A^*A.


Consequences

The concept of normality is important because normal matrices are precisely those to which the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
applies: The diagonal entries of are the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of , and the columns of are the
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of . The matching eigenvalues in come in the same order as the eigenvectors are ordered as columns of . Another way of stating the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
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 In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of . Phrased differently: a matrix is normal if and only if its
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s span and are pairwise orthogonal with respect to the standard inner product of . The spectral theorem for normal matrices is a special case of the more general
Schur decomposition In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper tri ...
which holds for all square matrices. Let be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, . If is normal, so is . But then 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: In general, the sum or product of two normal matrices need not be normal. However, the following holds: In this special case, the columns of are eigenvectors of both and and form an orthonormal basis in . This follows by combining the theorems that, over an algebraically closed field,
commuting matrices In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the s ...
are
simultaneously triangularizable In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
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 be a complex matrix. Then the following are equivalent: # is normal. # is
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
by a unitary matrix. # There exists a set of eigenvectors of which forms an orthonormal basis for . # \left\, A \mathbf \right\, = \left\, A^* \mathbf \right\, for every . # The
Frobenius norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
of can be computed by the eigenvalues of : \operatorname \left(A^* A\right) = \sum_j \left, \lambda_j \^2 . # The
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
part and
skew-Hermitian __NOTOC__ In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relatio ...
part of commute. # is a polynomial (of degree ) in .Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that \overline = P(\lambda_j), where \lambda_j are the eigenvalues of A. # for some unitary matrix . # and commute, where we have the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
with a unitary matrix and some positive semidefinite matrix . # commutes with some normal matrix with distinct eigenvalues. # for all where has
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
and eigenvalues . 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.


Normal matrix analogy

It is occasionally useful (but sometimes misleading) to think of the relationships of special kinds of normal matrices as analogous to the relationships of the corresponding type of complex numbers of which their eigenvalues are composed. This is because any function of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition VD V^* is VD^*V^*, where D is the diagonal matrix of eigenvalues. Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on each corresponding pair of eigenvalues. * The
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
is analogous to the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. *
Unitary matrices In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose ...
are analogous to
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. *
Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
are analogous to
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. * Hermitian
positive definite matrices In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
are analogous to
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
. * Skew Hermitian matrices are analogous to purely
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
s. *
Invertible matrices In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
are analogous to non-zero
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. * The inverse of a matrix has each eigenvalue inverted. * A uniform scaling matrix is analogous to a constant number. * In particular, the
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
is analogous to 0, and * the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
matrix is analogous to 1. * An
idempotent matrix In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^2 = A. For this product A^2 to be defined, A must necessarily be a square matrix. Viewed thi ...
is an orthogonal projection with each eigenvalue either 0 or 1. * A normal
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
has eigenvalues \pm 1. As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mapping a + bi \mapsto \begin a & b \\ -b & a \end = a\, \begin 1 & 0 \\ 0 & 1 \end + b\, \begin 0 & 1 \\ -1 & 0 \end\,. which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.


See also

*
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
* Least-squares normal matrix


Notes


Citations


Sources

* . * {{Matrix classes Matrices ja:正規作用素