Outer product

The dyadic product ( briefly also dyad of Greek δύας, Dyas " duality " ) or tensor product is in mathematics, a special product of two vectors. The result of a dyadic product is a matrix or a tensor of second order, with the rank one. The dyadic product can be viewed as a special case of a one-column matrix product with a single-line matrix; in this case it also corresponds to the Kronecker product of two matrices. To emphasize the contrast to the inner product ( scalar product ), the dyadic product is sometimes called the outer product, which term is also used for the cross product and the roof product.

The concept of the dyadic product goes back to the American physicist Josiah Willard Gibbs, who first formulated it in 1881 as part of its vector analysis.

• 5.1 scalar
• 5.2 tensor

Definition

The dyadic product is a combination of two real vectors and the shape

The result being a matrix. Each entry of the resulting matrix is calculated from the vectors and

As the product of the elements and. If one interprets the first vector as a column matrix and the second vector as a single-line matrix, so can the dyadic product using

Represented as a matrix product, wherein the vector is transposed to. The dyadic product can be viewed as a special case of the Kronecker product of a single column with a single-line matrix.

Examples

If and, then is the dyadic product of and

Each column in the matrix is therefore a multiple of, and each row is a multiple of. As trivial examples are each zero matrix, the dyadic product of zero vectors and each one matrix, the dyadic product of vectors corresponding to one of appropriate size:

Properties

The following properties of the dyadic product follow directly from the properties of the matrix multiplication.

Commutativity

For the transpose of the dyadic product of two vectors and is

The dyadic product is so if and commutative, which means that it applies

If the result matrix is symmetric. This is exactly the case when one of the two vectors is a multiple of the other vector, that is, when there are a number, or so is considered. If one of the vectors is a zero vector, then applies particularly to all

The resulting matrix then is the zero matrix.

Distributivity

With the vector addition of the dyadic product is always distributive, which means it applies to all and

And for all and accordingly

Furthermore, the dyadic product is compatible with the scalar multiplication, ie for and and applies

Rank -one matrices

The dyadic product of two vectors and results, unless one of the two vectors is the zero vector, a rank -one matrix, ie

Conversely, each rank -one matrix can be represented as a dyadic product of two vectors. For the spectral norm and the Frobenius norm of a dyadic product

The Euclidean norm of the vector. In addition to the zero matrix rank -one matrices are the only matrices for the match these two standards.

References to other products

Scalar product

If one forms the contrary the product of a row vector by a column vector, the result is the standard scalar product of two vectors is given by

Wherein the result is a real number. The standard scalar product of two vectors is equal to the trace ( the sum of the diagonal elements ) of their dyadic product, ie

Further, the matrix is ​​then nilpotent exactly when the two vectors are orthogonal, that is,

If alternate row and column vectors of appropriate size, several vectors can be multiplied together. Due to the associativity of matrix multiplication, one obtains the identities

And

A scalar product is also called inner product, so the dyadic product is sometimes referred to as the outer product. This duality is used in the Bra- Ket notation of quantum mechanics, where an inner product is listed by and by an outer product.

Tensor

The vector space which is spanned by the dyadic product of vectors, the Tensorproduktraum

This space is isomorphic to the space of all matrices. Each matrix can therefore be represented as a linear combination of dyadic products of vectors, ie

Where, and are. By a suitable choice of vectors and a ranked barrier, a low- rank approximation of a matrix can be reached in this way, making numerical calculations can be accelerated with very large matrices.

Use

In many applications, a dyadic product is not componentwise calculated, but first to stand and only then evaluated if it is multiplied by other terms. Multiplying the dyadic product with a vector one obtains a vector that is parallel to, as

Applies. The dyadic product of a unit vector with itself is a projection operator, since the matrix-vector product

Projects a given vector orthogonal to a line through the origin with direction vector. The reflection of a vector on a plane origin with unit normal vector is correspondingly as

Wherein the unit matrix. Such reflections are used for example in the Householder transformation.