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In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K (m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(A T): K (m,n) vec(A) = vec(A T) .
Comment: Every normal matrix A (that is, matrix for which =, where is a conjugate transpose) can be eigendecomposed. For a normal matrix A (and only for a normal matrix), the eigenvectors can also be made orthonormal ( V V ∗ = I {\displaystyle VV^{*}=I} ) and the eigendecomposition reads as A = V D V ∗ {\displaystyle A=VDV^{*}} .
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. [1] [2]In mathematics, a weighing matrix of order and weight is a matrix with entries from the set {,,} such that:
Gradient of the 2D function f(x, y) = xe −(x 2 + y 2) is plotted as arrows over the pseudocolor plot of the function.. Consider a room where the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z), independent of time.
The Rail Fence cipher is a form of transposition cipher that gets its name from the way in which it is encoded. In the rail fence cipher, the plaintext is written downwards and diagonally on successive "rails" of an imaginary fence, then moving up when we get to the bottom.
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, = for matrices A, B, and C of dimensions k×l, l×m, and m×n.
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of ...