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Similar to the one-dimensional case, an asterisk is used to represent the convolution operation. The number of dimensions in the given operation is reflected in the number of asterisks. For example, an M-dimensional convolution would be written with M asterisks. The following represents a M-dimensional convolution of discrete signals:
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.
One can consider multilinear functions, on an n×n matrix over a commutative ring K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and a i, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as = (, …,),
When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating ...
Verlet integration (French pronunciation:) is a numerical method used to integrate Newton's equations of motion. [1] It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics.
Cartesian velocity moments are based on these Cartesian moments. A Cartesian velocity moment v m p q μ γ {\displaystyle vm_{pq\mu \gamma }} is defined by v m p q μ γ = ∑ i = 2 i m a g e s ∑ x = 1 M ∑ y = 1 N U ( i , μ , γ ) C ( i , p , g ) P i x y {\displaystyle vm_{pq\mu \gamma }=\sum _{i=2}^{images}\sum _{x=1}^{M}\sum _{y=1}^{N}U ...
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m×n matrix and B is an n×p matrix, then their matrix product AB is the m×p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column ...
In analytical mechanics, the mass matrix is a symmetric matrix M that expresses the connection between the time derivative ˙ of the generalized coordinate vector q of a system and the kinetic energy T of that system, by the equation