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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.
[a] This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the ...
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation which directly varies from point to point in the domain of the function.