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Encyclopedia > Dyadic tensor

A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i.e. placing pairs of vectors side by side. In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ...

Each component of a dyadic tensor is a dyad. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.

As an example, let

$mathbf{A} = a mathbf{i} + b mathbf{j}$

and

$mathbf{X} = x mathbf{i} + y mathbf{j}$

be a pair of two-dimensional vectors. Then the juxtaposition of A and X is

$mathbf{A X} = a x mathbf{i i} + a y mathbf{i j} + b x mathbf{j i} + b y mathbf{j j}$ .

The identity dyadic tensor in three dimensions is

i i + j j + k k.

The dyadic tensor

j i − i j

is a 90° rotation operator in two dimensions. It can be dotted (from the left) with a vector to produce the rotation: This article concerns the rotation operator, as it appears in quantum mechanics. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

$(mathbf{j i} - mathbf{i j}) cdot (x mathbf{i} + y mathbf{j}) = x mathbf{j i} cdot mathbf{i} - x mathbf{i j} cdot mathbf{i} + y mathbf{j i} cdot mathbf{j} - y mathbf{i j} cdot mathbf{j} = -y mathbf{i} + x mathbf{j}.$

Dyadic tensor - TheBestLinks.com - Tensor, Fr:Tenseur dyadique, ... (248 words)
	A dyadic tensor is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i.e.
	The identity dyadic tensor in three dimensions is i i + j j + k k.
	The dyadic tensor j i - i j is a 90 degree rotation operator in two dimensions.

Dyadic tensor - Wikipedia, the free encyclopedia (137 words)
	A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i.e.
	Each component of a dyadic tensor is a dyad.
	A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.

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