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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.
Simply, in coordinates, the inner product is the product of a covector with an vector, yielding a matrix (a scalar), while the outer product is the product of an vector with a covector, yielding an matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension.
Block matrix operations; Cracovian product, defined as A ∧ B = B T A; Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product; Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry
The product cA of a number c (also called a scalar in this context) and a matrix A is computed by multiplying every entry of A by c: (), =, This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "inner product". For example:
In the special case v i = w i, the inner product is the square norm of the k-vector, given by the determinant of the Gramian matrix ( v i, v j ). This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on ().
The Frobenius inner product may be extended to a hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing B by its complex conjugate. The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the ...
Hadamard product (matrices) Hilbert–Schmidt inner product; Kronecker product; Matrix analysis; Matrix multiplication; Matrix norm; Tensor product of Hilbert spaces – the Frobenius inner product is the special case where the vector spaces are finite-dimensional real or complex vector spaces with the usual Euclidean inner product
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...