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The transformed data matrix Y is obtained from the original matrix X by centering and optionally standardizing the columns (the variables). Using the SVD, we can write Y = Σ k =1,... p d k u k v k T ;, where the u k are n -dimensional column vectors, the v k are p -dimensional column vectors, and the d k are a non-increasing sequence of non ...
For a quaternion q = a + bi + cj + dk, Hamilton used two projections: S q = a, for the scalar part of q, and V q = bi + cj + dk, the vector part. Using the modern terms cross product (×) and dot product (.), the quaternion product of two vectors p and q can be written pq = –p.q + p×q.
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
This correlation would also map a line determined by two points (a 1, b 1, c 1, d 1) and (a 2, b 2, c 2, d 2) to the line which is the intersection of the two planes with equations a 1 x + b 1 y + c 1 z + d 1 w = 0 and a 2 x + b 2 y + c 2 z + d 2 w = 0. The associated sesquilinear form for this correlation is: φ(u, x) = u H ⋅ x P = u 0 x 0 ...
It forms a loop in the first quadrant with a double point at the origin and asymptote + + =. It is symmetrical about the line y = x {\displaystyle y=x} . As such, the two intersect at the origin and at the point ( 3 a / 2 , 3 a / 2 ) {\displaystyle (3a/2,3a/2)} .
A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a computer. In the past, sometimes mechanical or electronic plotters were used. Graphs are a visual representation of the relationship between variables, which are very ...
If imagined as a parallelogram, with the origin for the vectors at 0, then signed area is the determinant of the vectors' Cartesian coordinates (a x b y − b x a y). [21] The cross product a × b is orthogonal to the bivector a ∧ b. In three dimensions all bivectors can be generated by the exterior product of two vectors.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.