|
In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...
In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
- AT = A
which implies that A is a square matrix. The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then For the square matrix section, see square matrix. ...
In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...
- aij = aji
for all indices i and j.
The following 3-by-3 matrix is symmetric:
 Any diagonal matrix is symmetric, since all its off-diagonal entries are zero. In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...
A matrix is called skew-symmetric if its transpose is the same as its negative. In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
Properties One of the basic theorems concerning such matrices is the finite-dimensional spectral theorem, which says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: to every symmetric real matrix A there exists a real orthogonal matrix Q such that D = QTAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ...
Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...
In mathematics, an orthonormal basis of an inner product space V(i. ...
Another way of stating the spectral theorem is that the eigenvectors of a symmetric matrix are orthogonal. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the above diagonal matrix D, and therefore D is uniquely determined by A, up to the order of its entries.) Essentially, the property of symmetry of real matrices corresponds to the property of being Hermitian for complex matrices. A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for...
Every square real matrix X can be written in a unique way as the sum of a symmetric and a skew-symmetric matrix. This is done in the following way: In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
 (This is true more generally for every square matrix X with entries from any field whose characteristic is different from 2.) This article presents the essential definitions. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i.e. if AB = BA. Two real symmetric matrices commute if and only if they have the same eigenspaces. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
Any matrix congruent to a symmetric matrix is again symmetric: if X is a symmetric matrix then so is AXAT for any matrix A. In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
Denote with <,> the standard inner product on Rn. The real n-by-n matrix A is symmetric if and only if // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...
 . Using the Jordan normal form, 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. (Bosch, 1986) In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M...
Occurrence Symmetric real n-by-n matrices appear as the Hessian of twice continuously differentiable functions of n real variables. In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...
Every quadratic form q on Rn can be uniquely written in the form q(x) = xTAx with a symmetric n-by-n matrix A. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of Rn, "looks like" In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
 with real numbers λi. This considerably simplifies the study of quadratic forms, as well as the study of the level sets {x : q(x) = 1} which are generalizations of conic sections. Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...
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, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...
See also Other types of symmetry or pattern in square matrices have special names; see for example: Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
See also symmetry in mathematics. In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is shifted one element to the right relative to the preceding row vector. ...
In linear algebra, a Hankel matrix, named after Hermann Hankel, is a square matrix with constant (positive sloping) skew-diagonals, e. ...
In the mathematical discipline of 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. ...
Symmetry in mathematics occurs not only in geometry, but also in other branches of mathematics. ...
References - A. J. Bosch (1986). "The factorization of a square matrix into two symmetric matrices". American Mathematical Monthly 93: 462-464.
|
Feeds |
Contact