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In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not depend on the ...
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. [1] [2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For ...
The measure gives rise to an (,)-sized similarity matrix for a set of n points, where the entry (,) in the matrix can be simply the (reciprocal of the) Euclidean distance between and , or it can be a more complex measure of distance such as the Gaussian ‖ ‖ /. [5]
The dot product satisfies the properties [1] It is symmetric in x and y: x ⋅ y = y ⋅ x. It is linear in its first argument: (ax 1 + bx 2) ⋅ y = a(x 1 ⋅ y) + b(x 2 ⋅ y) for any scalars a, b, and vectors x 1, x 2, and y. It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0.
Vectors can be specified using either ordered pair notation (a subset of ordered set notation using only two components), or matrix notation, as with rectangular coordinates. In these forms, the first component of the vector is r (instead of v 1 ), and the second component is θ (instead of v 2 ).
where {e 1 ∧ e 2, e 3 ∧ e 1, e 2 ∧ e 3} is the basis for the three-dimensional space ⋀ 2 (R 3). The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a bivector .
The tensor product of two vectors is a second-order tensor, although this has no obvious directional interpretation by itself. The previous idea can be continued: if T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention ...
In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used.