Table of Contents
What is a tensor in maths?
Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor.
What is tensor dot product?
Tensordot (also known as tensor contraction) sums the product of elements from a and b over the indices specified by axes . Example 3: When a and b are matrices (order 2), the case axes=0 gives the outer product, a tensor of order 4. Example 4: Suppose that a i j k and b l m n represent two tensors of order 3.
What do you mean by the contraction of a tensor?
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. The result is another tensor with order reduced by 2. Tensor contraction can be seen as a generalization of the trace.
How do you write a tensor product?
If x, y are vectors of length M and N, respectively, their tensor product x⊗y is defined as the M ×N-matrix defined by (x ⊗ y)ij = xiyj. In other words, x ⊗ y = xyT . In particular x ⊗ y is a matrix of rank 1, which means that most matrices cannot be written as tensor products.
How do you read tensor notation?
Tensor notation introduces one simple operational rule. It is to automatically sum any index appearing twice from 1 to 3. As such, aibj a i b j is simply the product of two vector components, the ith component of the a vector with the jth component of the b vector.
What is inner product of vectors?
An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar. More precisely, for a real vector space, an inner product satisfies the following four properties.
What can the tensor product be extended to?
More generally, the tensor product can be extended to other categories of mathematical objects in addition to vector spaces, such as to matrices, tensors, algebras, topological vector spaces, and modules.
How do you find the tensor product of two vectors?
The first is a vector (v,w) ( v, w) in the direct sum V ⊕W V ⊕ W (this is the same as their direct product V ×W V × W ); the second is a vector v ⊗w v ⊗ w in the tensor product V ⊗W V ⊗ W. And that’s it! Forming the tensor product v⊗w v ⊗ w of two vectors is a lot like forming the Cartesian product of two sets X×Y X × Y.
How do you write a dot product in tensor notation?
A dot product of vectors\\({\\bf a}\\) and \\({\\bf b}\\) is written in tensor notation simply as \\( a_i b_i \\). The summation from 1 to 3 is implied because the subscript ( \\( i \\) in this case ) appears twice ( on \\( a \\) and \\( b \\) ).
How do you sum a vector in tensor notation?
Summation Convention Tensor notation introduces one simple operational rule. It is to automatically sum any index appearing twice from 1 to 3. As such, aibj a i b j is simply the product of two vector components, the i th component of the a a vector with the j th component of the b b vector.