In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, a symmetric matrix is a
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 ...
that is equal to its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
. So if
denotes the entry in the
th row and
th column then
for all indices
and
Every square
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 m ...
is symmetric, since all off-diagonal elements are zero. Similarly in
characteristic different from 2, each diagonal element of a
skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
must be zero, since each is its own negative.
In linear algebra, a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
symmetric matrix represents a
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
represented in an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
over a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
. The corresponding object for 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 ...
inner product space is a
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 ...
with complex-valued entries, which is equal to 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 ...
. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Example
The following
matrix is symmetric:
Properties
Basic properties
* The sum and difference of two symmetric matrices is symmetric.
* This is not always true for the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
: given symmetric matrices
and
, then
is symmetric if and only if
and
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 ...
, i.e., if
.
* For any integer
,
is symmetric if
is symmetric.
* If
exists, it is symmetric if and only if
is symmetric.
* Rank of a symmetric matrix
is equal to the number of non-zero eigenvalues of
.
Decomposition into symmetric and skew-symmetric
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let
denote the space of
matrices. If
denotes the space of
symmetric matrices and
the space of
skew-symmetric matrices then
and
, i.e.
where
denotes the
direct sum. Let
then
Notice that
and
. This is true for every
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 ...
with entries from any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
whose
characteristic is different from 2.
A symmetric
matrix is determined by
scalars (the number of entries on or above the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
). Similarly, a
skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
is determined by
scalars (the number of entries above the main diagonal).
Matrix congruent to a symmetric matrix
Any matrix
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to a symmetric matrix is again symmetric: if
is a symmetric matrix, then so is
for any matrix
.
Symmetry implies normality
A (real-valued) symmetric matrix is necessarily a
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
.
Real symmetric matrices
Denote by
the standard
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on
. The real
matrix
is symmetric if and only if
Since this definition is independent of the choice of
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
, symmetry is a property that depends only on the
linear operator A and a choice of
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. This characterization of symmetry is useful, for example, in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, for each
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
may be endowed with an inner product, giving rise to what is called a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. Another area where this formulation is used is in
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.
The finite-dimensional
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 ...
says that any symmetric matrix whose entries are
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
can be
diagonalized
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 an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
. More explicitly: For every real symmetric matrix
there exists a real orthogonal matrix
such that
is 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 m ...
. Every real symmetric matrix is thus,
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
choice of an
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, ...
, a diagonal matrix.
If
and
are
real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of
such that every element of the basis is an
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 ...
for both
and
.
Every real symmetric matrix is
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 therefore all its
eigenvalues
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 ...
are real. (In fact, the eigenvalues are the entries in the diagonal matrix
(above), and therefore
is uniquely determined by
up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for complex matrices.
Complex symmetric matrices
A complex symmetric matrix can be 'diagonalized' using a
unitary matrix
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 is ...
: thus if
is a complex symmetric matrix, there is a unitary matrix
such that
is a real diagonal matrix with non-negative entries. This result is referred to as the Autonne–Takagi factorization. It was originally proved by
Léon Autonne (1915) and
Teiji Takagi
Teiji Takagi (高木 貞治 ''Takagi Teiji'', April 21, 1875 – February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere-differentiabl ...
(1925) and rediscovered with different proofs by several other mathematicians. In fact, the matrix
is Hermitian and
positive semi-definite, so there is a unitary matrix
such that
is diagonal with non-negative real entries. Thus
is complex symmetric with
real. Writing
with
and
real symmetric matrices,
. Thus
. Since
and
commute, there is a real orthogonal matrix
such that both
and
are diagonal. Setting
(a unitary matrix), the matrix
is complex diagonal. Pre-multiplying
by a suitable diagonal unitary matrix (which preserves unitarity of
), the diagonal entries of
can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as
. The matrix we seek is simply given by
. Clearly
as desired, so we make the modification
. Since their squares are the eigenvalues of
, they coincide with the
singular value 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 ...
s of
. (Note, about the eigen-decomposition of a complex symmetric matrix
, the Jordan normal form of
may not be diagonal, therefore
may not be diagonalized by any similarity transformation.)
Decomposition
Using the
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.
Every real
non-singular matrix
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 ...
can be uniquely factored as the product of an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
and a symmetric
positive definite matrix
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 co ...
, which is called a
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 ...
. Singular matrices can also be factored, but not uniquely.
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
states that every real positive-definite symmetric matrix
is a product of a lower-triangular matrix
and its transpose,
If the matrix is symmetric indefinite, it may be still decomposed as
where
is a permutation matrix (arising from the need to
pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
),
a lower unit triangular matrix, and
is a direct sum of symmetric
and
blocks, which is called Bunch–Kaufman decomposition
A general (complex) symmetric matrix may be
defective
Defective may refer to::
*Defective matrix, in algebra
*Defective verb, in linguistics
*Defective, or ''haser'', in Hebrew orthography, a spelling variant that does not include mater lectionis
*Something presenting an anomaly, such as a product de ...
and thus not be
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 ...
. If
is diagonalizable it may be decomposed as
where
is an orthogonal matrix
, and
is a diagonal matrix of the eigenvalues of
. In the special case that
is real symmetric, then
and
are also real. To see orthogonality, suppose
and
are eigenvectors corresponding to distinct eigenvalues
,
. Then
Since
and
are distinct, we have
.
Hessian
Symmetric
matrices of real functions appear as the
Hessians of twice differentiable functions of
real variables (the continuity of the second derivative is not needed, despite common belief to the opposite
).
Every
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
on
can be uniquely written in the form
with a symmetric
matrix
. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of
, "looks like"
with real numbers
. This considerably simplifies the study of quadratic forms, as well as the study of the level sets
which are generalizations of
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s.
This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
.
Symmetrizable matrix
An
matrix
is said to be symmetrizable if there exists an invertible
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 m ...
and symmetric matrix
such that
The transpose of a symmetrizable matrix is symmetrizable, since
and
is symmetric. A matrix
is symmetrizable if and only if the following conditions are met:
#
implies
for all
#
for any finite sequence
See also
Other types of
symmetry
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 definit ...
or pattern in square matrices have special names; see for example:
*
Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
(also called ''antisymmetric'' or ''antimetric'')
*
Centrosymmetric matrix
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an ''n''×''n'' matrix ''A'' = 'A'i'',''j''is centrosymmetric when its entries satisfy
: ...
*
Circulant matrix
In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplit ...
*
Covariance matrix
*
Coxeter matrix
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
*
GCD matrix
In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix.
Definition
Let S=(x_1, x_2,\ldots, x_n) be a list of positive integers. Then the n\times n matrix (S) having the greatest common divisor
In ...
*
Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...
*
Hilbert matrix In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions
: H_ = \frac.
For example, this is the 5 × 5 Hilbert matrix:
: H = \begin
1 & \frac & \frac & \frac & \frac \\
\frac & \frac & \f ...
*
Persymmetric matrix In mathematics, persymmetric matrix may refer to:
# a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or
# a square matrix such that the values on each line perpendicular to the main diagonal are the same for a ...
*
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadra ...
*
Toeplitz matrix In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:
:\qquad\begin
a & b ...
*
Transpositions matrix Transpositions matrix (Tr matrix) is square n \times n matrix, n=2^, m \in N , which elements are obtained from the elements of given n-dimensional vector X=(x_i)_ as follows: Tr_ = x_, where \oplus denotes operation "bitwise Exclusive or" (XOR). Th ...
See also
symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
Given a structured obje ...
.
Notes
References
*
External links
*
A brief introduction and proof of eigenvalue properties of the real symmetric matrixHow to implement a Symmetric Matrix in C++
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Matrices