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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. For such matrices, the half-vectorization is sometimes more useful than the ...
Floating-point fused multiply-add instructions are introduced in x86 as two instruction set extensions, "FMA3" and "FMA4", both of which build on top of AVX to provide a set of scalar/vector instructions using the xmm/ymm/zmm vector registers. FMA3 defines a set of 3-operand fused-multiply-add instructions that take three input operands and ...
The components of a vector are often represented arranged in a column. By contrast, a covector has components that transform like the reference axes. It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector.
Function rank is an important concept to array programming languages in general, by analogy to tensor rank in mathematics: functions that operate on data may be classified by the number of dimensions they act on. Ordinary multiplication, for example, is a scalar ranked function because it operates on zero-dimensional data (individual numbers).
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
Let us define scalar and vector such that quaternion = (,). Note that the canonical way to rotate a three-dimensional vector v → {\displaystyle {\vec {v}}} by a quaternion q {\displaystyle q} defining an Euler rotation is via the formula
Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its ...
The term "component" of a vector is ambiguous: it could refer to: a specific coordinate of the vector such as a z (a scalar), and similarly for x and y, or; the coordinate scalar-multiplying the corresponding basis vector, in which case the "y-component" of a is a y e y (a vector), and similarly for x and z.