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In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector.
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.
[4]: 7 Furthermore, as noted in the above formula, tr(A ⊤ B) = tr(B ⊤ A). These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call tr(A ⊤ B) the Frobenius inner product of A and B. This is a natural inner product on the vector space of all real matrices of fixed dimensions.
Automatic vectorization, a compiler optimization that transforms loops to vector operations Image tracing , the creation of vector from raster graphics Word embedding , mapping words to vectors, in natural language processing
In the case of column vectors, the Kronecker product can be viewed as a form of vectorization (or flattening) of the outer product. In particular, for two column vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } , we can write:
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.