Children: Matrices
The Nekrasov matrix is a concept that arises in the context of mathematical physics, particularly in the study of supersymmetric gauge theories and their connections to algebraic geometry and integrable systems. It is named after the Russian mathematician Nikita Nekrasov, who contributed significantly to the field.
The term "next-generation matrix" can refer to various concepts depending on the context in which it is used. However, it is not a widely recognized term in scientific literature or popular technologies as of my last update in October 2023. Below are a few possible interpretations based on the context of matrices in technology and computing: 1. **Quantum Computing**: In quantum computing, matrices play a crucial role, especially in representing quantum states and operations.
A **nilpotent matrix** is a square matrix \( A \) such that there exists some positive integer \( k \) for which the matrix raised to the power of \( k \) equals the zero matrix.
A nonnegative matrix is a type of matrix in which all the elements are greater than or equal to zero.
In linear algebra, a **normal matrix** is a type of matrix that commutes with its own conjugate transpose. Specifically, a square matrix \( A \) is defined as normal if it satisfies the condition: \[ AA^* = A^*A \] where \( A^* \) denotes the conjugate transpose (or Hermitian transpose) of matrix \( A \).
Orbital overlap refers to the phenomenon that occurs when atomic orbitals from two adjacent atoms come close enough to each other that their electron clouds can interact. This overlap is crucial for the formation of chemical bonds, such as covalent bonds, in which electrons are shared between atoms. In covalent bonding, the greater the overlap of the atomic orbitals, the stronger the bond that is formed.
An orthogonal matrix is a square matrix \( A \) whose rows and columns are orthogonal unit vectors. This means that: 1. The dot product of any two different rows (or columns) is zero, indicating that they are orthogonal (perpendicular). 2. The dot product of a row (or column) with itself is one, indicating that the vectors are normalized.
An orthostochastic matrix is a mathematical construct that arises in the context of stochastic processes and linear algebra. Specifically, it is a type of matrix associated with stochastic transformations, preserving certain probabilistic properties. A matrix \( A \) is termed orthostochastic if it satisfies the following conditions: 1. **Non-negativity:** All entries of \( A \) are non-negative, meaning \( a_{ij} \geq 0 \) for all entries \( i, j \).
A \( P \)-matrix is a mathematical concept that arises in the study of matrix theory and game theory. Specifically, a matrix \( A \) is called a \( P \)-matrix if all its leading principal minors are positive.
A Packed Storage Matrix (PSM) is a data structure used to efficiently store and manipulate sparse matrices, which contain a significant number of zero elements. Instead of storing all matrix elements in a standard two-dimensional array (which would consume a lot of memory for large matrices), a packed storage format only saves the non-zero entries along with any necessary information to reconstruct the matrix.
Paley construction refers to a method of constructing finite groups from properties of finite abelian groups, particularly using characters and representation theory. Named after the mathematician Arthur Paley, this construction involves building groups that have specific properties, often relating to their order or symmetry. A notable application of Paley's work is in the construction of Paley graphs, which are a specific type of graph used in number theory and combinatorial design.
The Pauli matrices are a set of three \(2 \times 2\) complex matrices that are widely used in quantum mechanics, particularly in the study of spin and other quantum two-level systems (qubits). They are named after the physicist Wolfgang Pauli.
A perfect matrix, also known as a perfect matching matrix, is a concept from graph theory, rather than a standard term in linear algebra. In the context of bipartite graphs, a perfect matching is a set of edges that pairs up all vertices from one set to the other without any overlaps. For example, consider a bipartite graph \( G = (U, V, E) \) where \( U \) and \( V \) are disjoint sets of vertices.
A permutation matrix is a special type of square binary matrix that is used to represent a permutation of a finite set. Specifically, it is an \( n \times n \) matrix that contains exactly one entry of 1 in each row and each column, and all other entries are 0.
A persymmetric matrix, also known as a symmetric Toeplitz matrix, is a special type of square matrix that exhibits symmetry in a specific manner. An \( n \times n \) matrix \( A \) is defined as persymmetric if it satisfies the condition: \[ A[i, j] = A[n-j+1, n-i+1] \] for all valid indices \( i \) and \( j \).
The Plücker matrix is a mathematical construct used in projective geometry and algebraic geometry, particularly in the context of analyzing lines in three-dimensional space. It is named after Julius Plücker, a 19th-century mathematician who contributed significantly to the field. In the context of lines in \(\mathbb{R}^3\), a line can be represented by a pair of points or by a direction vector along with a point through which the line passes.
A **polyconvex function** is a specific type of function commonly used in the field of calculus of variations and optimization, particularly in the study of vector-valued functions and elasticity theory. The concept is related to the notion of convexity, which involves the shape and properties of functions in relation to their inputs.
A polynomial matrix is a matrix whose entries are polynomials. In other words, each element of the matrix is a polynomial function of one or more variables. Polynomial matrices are used in various areas of mathematics and applied sciences, including control theory, systems theory, and algebra.
A projection matrix is a square matrix that transforms a vector into its projection onto a subspace. In the context of linear algebra, projections are used to reduce the dimensionality of data or to find the closest point in a subspace to a given vector. ### Key Properties of Projection Matrices: 1. **Idempotent**: A matrix \( P \) is a projection matrix if \( P^2 = P \).