Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
Diagonal matrixIn 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 matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size).
Adjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A). It is also occasionally known as adjunct matrix, or "adjoint", though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: where I is the identity matrix of the same size as A.
Locally convex topological vector spaceIn functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family.
Definite matrixIn mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative).
Triangular matrixIn 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 zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.
Rank–nullity theoremThe rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the of f) and the nullity of f (the dimension of the kernel of f). It follows that for linear transformations of vector spaces of finite dimension, either injectivity or surjectivity implies bijectivity.
Linear subspaceIn mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K.
Sequence spaceIn functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.
LF-spaceIn mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces. If each of the bonding maps is an embedding of TVSs then the LF-space is called a strict LF-space.
Matrix ringIn abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication . The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra.
Pivot elementThe pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error.
