Linear map in the context of Self-adjoint operator

In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.

Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.

Linear algebra is the branch of mathematics concerning linear equations such as

linear maps such as

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

where ${\displaystyle V_{1},\ldots ,V_{n}}$ (${\displaystyle n\in \mathbb {Z} _{\geq 0}}$) and ${\displaystyle W}$ are vector spaces (or modules over a commutative ring), with the following property: for each ${\displaystyle i}$, if all of the variables but ${\displaystyle v_{i}}$ are held constant, then ${\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}$ is a linear function of ${\displaystyle v_{i}}$. One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of ${\displaystyle 2^{2}}$.

Planar projections are the subset of 3D graphical projections constructed by linearly mapping points in three-dimensional space to points on a two-dimensional projection plane. The projected point on the plane is chosen such that it is collinear with the corresponding three-dimensional point and the centre of projection. The lines connecting these points are commonly referred to as projectors.

The centre of projection can be thought of as the location of the observer, while the plane of projection is the surface on which the two dimensional projected image of the scene is recorded or from which it is viewed (e.g., photographic negative, photographic print, computer monitor). When the centre of projection is at a finite distance from the projection plane, a perspective projection is obtained. When the centre of projection is at infinity, all the projectors are parallel, and the corresponding subset of planar projections are referred to as parallel projections.

In mathematics, a transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X.Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations.

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.

The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra.

In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.

Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering.Computing matrix products is a central operation in all computational applications of linear algebra.

Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics.

In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples)

The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} ^{n}}$.Such operators often preserve properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

In algebra, an additive map, ${\displaystyle \mathbb {Z} }$-linear map or additive function is a function ${\displaystyle f}$ that preserves the addition operation:$${\displaystyle f(x+y)=f(x)+f(y)}$$for every pair of elements ${\displaystyle x}$ and ${\displaystyle y}$ in the domain of ⁠${\displaystyle f}$⁠. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial.

More formally, an additive map is a ${\displaystyle \mathbb {Z} }$-module homomorphism. Since an abelian group is a ${\displaystyle \mathbb {Z} }$-module, it may be defined as a group homomorphism between abelian groups.

A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit F(x) when a linear combination of signals ax₁(t) + bx₂(t) is applied to it is equal to the linear combination of the outputs due to the signals x₁(t) and x₂(t) applied separately:

It is called a linear circuit because the output voltage and current of such a circuit are linear functions of its input voltage and current. This kind of linearity is not the same as that of straight-line graphs.

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector ${\displaystyle \mathbf {v} }$ of a linear transformation ${\displaystyle T}$ is scaled by a constant factor ${\displaystyle \lambda }$ when the linear transformation is applied to it: ${\displaystyle T\mathbf {v} =\lambda \mathbf {v} }$. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor ${\displaystyle \lambda }$ (possibly a negative or complex number).

Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed.

Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.

Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing.

The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified.