12 Facts About Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.

Dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar a,.

Dot product fulfills the following properties if a, b, and c are real vectors and r is a scalar.

The dot product is a scalar in this sense, given by the formula, independent of the coordinate system.

When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H:.

The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector.

However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in a The dot product is not symmetric, since.

Complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics.

An inner Dot product space is a normed vector space, and the inner Dot product of a vector with itself is real and positive-definite.

Dot product is defined for vectors that have a finite number of entries.

Inner Dot product between a tensor of order n and a tensor of order m is a tensor of order, see Tensor contraction for details.