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Query: What is a matrix in algebra?
In mathematics, a matrix (plural: matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns:
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, formula_61 is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).
The two propositions above imply that the algebra generated by elements of a Manin matrix is a bialgebra coacting on the polynomial algebra. If one does not impose other relations ones get algebra of non-commutative endomorphisms of the polynomial algebra.
In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra, especially the theory of semigroups.
In mathematics, a quaternion algebra over a field "F" is a central simple algebra "A" over "F" that has dimension 4 over "F". Every quaternion algebra becomes the matrix algebra by "extending scalars" (equivalently, tensoring with a field extension), i.e. for a suitable field extension "K" of "F", formula_1 is isomorphic to the 2×2 matrix algebra over "K".
A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix formula_1 and the all-ones matrix formula_2.
The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.
In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the Boolean algebra has just two elements {0,1} the Boolean matrix is called a logical matrix.
The matrix associated to an element "x" of "F" can also be used to give other, equivalent descriptions of algebraic integers. An element "x" of "F" is an algebraic integer if and only if the characteristic polynomial "p" of the matrix "A" associated to "x" is a monic polynomial with integer coefficients. Suppose that the matrix "A" that represents an element "x" has integer entries in some basis "e". By the Cayley–Hamilton theorem, "p"("A") = 0, and it follows that "p"("x") = 0, so that "x" is an algebraic integer. Conversely, if "x" is an element of "F" which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix "A". In this case it can be proven that "A" is an integer matrix in a suitable basis of "F". Note that the property of being an algebraic integer is "defined" in a way that is independent of a choice of a basis in "F".
More formally, the group algebra of the Heisenberg group over its field of scalars "K", written , has center , so rather than simply thinking of the group algebra as an algebra over the field , one may think of it as an algebra over the commutative algebra . As the center of a matrix algebra or operator algebra is the scalar matrices, a -structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of -algebras , which is the formal way of saying that it sends the center to a chosen scale.
Any Lie group defines an associated real Lie algebra formula_93. The definition in general is somewhat technical, but in the case of a real matrix group "G", it can be formulated via the exponential map, or the matrix exponential. The Lie algebra formula_8 of "G" may be computed as
The Lie bracket of formula_8 is given by the commutator of matrices, formula_97. The following are examples of Lie algebras of matrix Lie groups:
Given a vector space "V", let formula_69 denote the Lie algebra consisting of all linear endomorphisms of "V", with bracket given by formula_97. A "representation" of a Lie algebra formula_8 on "V" is a Lie algebra homomorphism
and therefore isomorphic (since "α" is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cℓ(C) is 2. What we have then is the following table:
Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix "C". In order to classify Kac–Moody algebras, it is enough to consider the case of an "indecomposable" matrix "C", that is, assume that there is no decomposition of the set of indices "I" into a disjoint union of non-empty subsets "I" and "I" such that "C" = 0 for all "i" in "I" and "j" in "I". Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:
The Lorentz group is a Lie group and has as such a Lie algebra, The Lie algebra is a vector space of matrices that can be said to model the group near the identity. It is endowed with a multiplication operation, the Lie bracket. With it, the product in the group can near the identity be expressed in Lie algebraic terms (but not in a particularly simple way). The link between the (matrix) Lie algebra and the (matrix) Lie group is the matrix exponential. It is one-to-one near the identity in the group.
"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac-Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." - A.J. Coleman
The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix formula_26.(Note that formula_27 is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows:
The projection matrix corresponding to a linear model is symmetric and idempotent, that is, formula_33. However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent.
Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix population models are a specific type of population model that uses matrix algebra. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations.
Representation theory is a branch of mathematics that studies abstract algebraic structures by "representing" their elements as linear transformations of vector spaces, and studies 
modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by "representing" their elements as linear transformations of vector spaces, and studies
modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
In algebraic graph theory, the adjacency algebra of a graph "G" is the algebra of polynomials in the adjacency matrix "A"("G") of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of "A".
In the quantization for an odd number 2"k"+1 of dimensions, the matrices "P" may be introduced as above for "i" = 1,2...,2"k", and the following matrix may be adjoined to the system:
so that the Clifford relations still hold. This adjunction has no effect on the algebra "A" of matrices generated by the "P", since in either case "A" is still a complete matrix algebra of the same dimension. Thus "A", which is a complete 2×2 matrix algebra, is not the Clifford algebra, which is an algebra of dimension 2×2×2. Rather "A" is the quotient of the Clifford algebra by a certain ideal.
If all of the entries on the main diagonal of a (upper or lower) triangular matrix are 0, the matrix is called strictly (upper or lower) triangular. All strictly triangular matrices are nilpotent, and the set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted formula_5 This algebra is the derived Lie algebra of formula_6, the Lie algebra of all upper triangular matrices; in symbols, formula_7 In addition, formula_8 is the Lie algebra of the Lie group of unitriangular matrices.
The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and nonassociative algebras. Given an integer "n", the ring of real square matrices of order "n" is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.
Many finite matroids may be represented by a matrix over a field "K", in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type over a field "F" may also be represented as an algebraic matroid over "F", by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist algebraic matroids that are not linear; indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero. However, if a matroid is algebraic over a field "F" of characteristic zero then it is linear over "F"("T") for some finite set of transcendentals "T" over "F" and over the algebraic closure of "F".