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In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m×n matrix, x designates a column vector (that is, n×1-matrix) of n variables x 1, x 2, ..., x n, and b is an m×1-column vector, then the matrix equation =
A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector. Again, an n -dimensional vector v {\displaystyle \mathbf {v} } can be specified in either of the following forms using matrices:
A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.
Vector projection, also known as vector resolute or vector component, a linear mapping producing a vector parallel to a second vector; Vector-valued function, a function that has a vector space as a codomain; Vectorization (mathematics), a linear transformation that converts a matrix into a column vector
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. [1] [2] The state-space method is characterized by the algebraization of general system theory, which makes it possible to use Kronecker vector-matrix structures.