Search results
Results from the WOW.Com Content Network
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, = + where is the directional derivative in the direction of multiplied by its magnitude.
The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(V) with V ⊗ V ∗ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products.
The identity mapping (or identity matrix), considered as a linear mapping or The trace or tensor contraction , considered as a mapping V ∗ ⊗ V → K {\displaystyle V^{*}\otimes V\to K} The map K → V ∗ ⊗ V {\displaystyle K\to V^{*}\otimes V} , representing scalar multiplication as a sum of outer products .
If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.