Search results
Results from the WOW.Com Content Network
In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions ...
These coordinate vectors form another vector space, which is isomorphic to the original vector space. A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.
The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics.
The dot product is the trace of the outer product. [5] Unlike the dot product, the outer product is not commutative. Multiplication of a vector by the matrix can be written in terms of the inner product, using the relation () = , .
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products: [1] (()) (()) = [] This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.
The dot product on is an example of a bilinear form which is also an inner product. [1] An example of a bilinear form that is not an inner product would be the four-vector product. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.