Linear algebra

concept in linear algebra

algebra over a field

vector space consisting of affine subsets

theorem

property of point sets in Euclidean spaces

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the two vectors are perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

coordinate system based on a triangle

complex square matrix such that its its conjugate transpose is equal to its negative

equivalence relation between matrices

particular type of mapping in linear algebra, also called transvection

relations between power sums and elementary symmetric functions

signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems

operation in abstract algebra

mathematical theorem about eigenvalues

field of mathematics about eigenvalues and eigenvectors of linear operators

map taking two vectors from a complex vector space and returning a complex number, which is linear in one variable and semilinear in another variable

group of linear transformations with determinant 1

In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.

polynomial of the elements of a matrix

vector space equipped with an alternating nondegenerate bilinear form

inequality which involves a linear function

construct in functional analysis

element of a basis for a function space

In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.

tool in linear algebra and matrix analysis

linear algebra

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

sequence of spaces in linear algebra

matrix that is symmetric about its center

quadratic form that is either greater then 0 except for 0 or less then 0 except for 0

thumb|In two dimensions, there are four orthants (called quadrants)

mathematical operator-valued measure of interest in quantum mechanics and functional analysis

basis of Euclidean space consisting of one-hot vectors

subset of a vector space closed under positive linear combinations

linear algebra

mathematics formula

construct for Hermitian matrices

function which maps multidimensional data to one dimension while preserving locality of the data points

a specialized flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes

mathematical theorem

linear algebra

regularization technique for ill-posed problems

сanonical form of matrices over a field

vector satisfying some of the criteria of an eigenvector

equivalence of matrices under row operations

algorithms for matrix decomposition

intersection of a line and a line can be the empty set, a point, or a line

specialized notation for multivariable calculus

in a vector space

Type of function in linear algebra

basis on a dual vector space canonically associated to a basis on the original vector space

vector that is annihilated by a quadratic form

A generalization of a basis to sets of possibly linearly dependent vectors which also satisfy the frame condition

regularization technique for ill-posed problems

rewriting of an expression into a simpler form

convex and balanced set

In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, \mathbf{x},\ \mathbf{y} \in \mathbb{R}^n, we say that \mathbf{x} weakly majorizes (or dominates) \mathbf{y} from below, commonly denoted \mathbf{x} \succ_w \mathbf{y}, when \sum_{i=1}^k x_i^{\downarrow} \geq \sum_{i=1}^k y_i^{\downarrow} for all k=1,\,\dots,\,n, where x_i^{\downarrow} denotes the ith largest entry of \mathbf{x}. If \mathbf{x}, \mathbf{y} further satisfy \sum_{i=1}^n x_i = \sum_{i=1}^n y_i, we say that \mathbf{x} majorizes (or dominates) \mathbf{y} , commonly denoted \mathbf{x} \succ \

inequalities in number theory and matrix theory