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48 relations: Additive inverse, Bounded operator, Cholesky decomposition, Closed range theorem, Complex number, Continuous functional calculus, Covariance matrix, Diagonal matrix, Diagonalizable matrix, Eigenvalues and eigenvectors, Formal power series, Hermitian adjoint, Holomorphic functional calculus, Idempotent matrix, Identity matrix, If and only if, Integer, Invertible matrix, Irrational number, Jordan normal form, Linear algebra, Logarithm of a matrix, Mathematics, Mathematics of Computation, Matrix (mathematics), Matrix function, Matrix multiplication, Methods of computing square roots, Minimal polynomial (linear algebra), Moore–Penrose inverse, Neumann series, Newton's method, Operator theory, Orthogonal matrix, Partial isometry, Positive-definite matrix, Pythagorean triple, Real number, Society for Industrial and Applied Mathematics, Spectral theorem, Square root, Square root of a 2 by 2 matrix, Sylvester's formula, Symmetric matrix, The Mathematical Gazette, Unbounded operator, Unitary matrix, Unscented transform.
Additive inverse
In mathematics, the additive inverse of a number is the number that, when added to, yields zero.
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Bounded operator
In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).
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Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations.
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Closed range theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Continuous functional calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.
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Covariance matrix
In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.
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Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Formal power series
In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.
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Hermitian adjoint
In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator.
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Holomorphic functional calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions.
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Idempotent matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.
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Identity matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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Irrational number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
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Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis.
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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Logarithm of a matrix
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematics of Computation
Mathematics of Computation is a bimonthly mathematics journal focused on computational mathematics.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix function
In mathematics, a matrix function is a function which maps a matrix to another matrix.
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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Methods of computing square roots
In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number.
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Minimal polynomial (linear algebra)
In linear algebra, the minimal polynomial of an matrix over a field is the monic polynomial over of least degree such that.
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Moore–Penrose inverse
In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix.
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Neumann series
A Neumann series is a mathematical series of the form where T is an operator.
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Newton's method
In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
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Operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.
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Orthogonal matrix
In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.
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Partial isometry
In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.
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Positive-definite matrix
In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.
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Pythagorean triple
A Pythagorean triple consists of three positive integers,, and, such that.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Society for Industrial and Applied Mathematics
The Society for Industrial and Applied Mathematics (SIAM) is an academic association dedicated to the use of mathematics in industry.
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Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
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Square root
In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.
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Square root of a 2 by 2 matrix
A square root of a 2 by 2 matrix M is another 2 by 2 matrix R such that M.
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Sylvester's formula
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function of a matrix as a polynomial in, in terms of the eigenvalues and eigenvectors of.
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
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The Mathematical Gazette
The Mathematical Gazette is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching.
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Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
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Unitary matrix
In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.
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Unscented transform
The unscented transform (UT) is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics.
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Redirects here:
Matrix square root, Sqrt of a matrix.
References
[1] https://en.wikipedia.org/wiki/Square_root_of_a_matrix
