In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix A is denoted det(A), det A, or .
The determinant of a 2 × 2 matrix is
and the determinant of a 3 × 3 matrix is
The determinant of an n × n matrix can be defined in several equivalent ways. The Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of n different entries, and the number of these summands is the factorial of n (the product of the n first positive integers). The Laplace expansion expresses the determinant of an n × n matrix as a linear combination of determinants of submatrices. Gaussian elimination expresses the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by a succession of elementary row operations.
Determinants can also be defined by some of their properties: the determinant is the unique function defined on the n × n matrices that has the four following properties. The determinant of the identity matrix is 1; the exchange of two rows (or of two columns) multiplies the determinant by −1; multiplying a row (or a column) by a number multiplies the determinant by this number; and adding to a row (or a column) a multiple of another row (or column) does not change the determinant.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or .
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.
We prove an identity relating the permanent of a rank 2 matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity
We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented H
2022
, , , , ,
Parkinson's disease (PD) is characterized by the selective degeneration of neuronal populations presumably due to pathogenic interactions between aging and predisposing factors such as increased level