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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Frobenius inner product is a binary operation that takes two
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension - same number of rows and columns, but are not restricted to be square matrices.


Definition

Given two
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
-valued ''n''×''m'' matrices A and B, written explicitly as : \mathbf = \,, \quad \mathbf = the Frobenius inner product is defined as, : \langle \mathbf, \mathbf \rangle_\mathrm =\sum_\overline B_ \, = \mathrm\left(\overline \mathbf\right) \equiv \mathrm\left(\mathbf^ \mathbf\right) where the overline denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, and \dagger denotes
Hermitian conjugate In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
. Explicitly this sum is :\begin \langle \mathbf, \mathbf \rangle_\mathrm = & \overline_ B_ + \overline_ B_ + \cdots + \overline_ B_ \\ & + \overline_ B_ + \overline_ B_ + \cdots + \overline_ B_ \\ & \vdots \\ & + \overline_ B_ + \overline_ B_ + \cdots + \overline_ B_ \\ \end The calculation is very similar to the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
, which in turn is an example of an inner product.


Relation to other products

If A and B are each
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued matrices, the Frobenius inner product is the sum of the entries of the Hadamard product. If the matrices are vectorised (i.e., converted into column vectors, denoted by " \mathrm(\cdot) "), then : \mathrm(\mathbf ) = ,\quad \mathrm(\mathbf ) = \,, \quad \overline^T\mathrm(\mathbf ) = Therefore : \langle \mathbf, \mathbf \rangle_\mathrm = \overline^T \mathrm(\mathbf) \, .


Properties

It is a
sesquilinear form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
, for four complex-valued matrices A, B, C, D, and two complex numbers ''a'' and ''b'': :\langle a\mathbf, b\mathbf \rangle_\mathrm = \overlineb\langle \mathbf, \mathbf \rangle_\mathrm :\langle \mathbf+\mathbf, \mathbf + \mathbf \rangle_\mathrm = \langle \mathbf, \mathbf \rangle_\mathrm + \langle \mathbf, \mathbf \rangle_\mathrm + \langle \mathbf, \mathbf \rangle_\mathrm + \langle \mathbf, \mathbf \rangle_\mathrm Also, exchanging the matrices amounts to complex conjugation: :\langle \mathbf, \mathbf \rangle_\mathrm = \overline For the same matrix, :\langle \mathbf, \mathbf \rangle_\mathrm \geq 0, and, :\langle \mathbf, \mathbf \rangle_\mathrm = 0 \Longleftrightarrow \mathbf = \mathbf.


Frobenius norm

The inner product induces the
Frobenius norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
:\, \mathbf\, _\mathrm = \sqrt \,.


Examples


Real-valued matrices

For two real-valued matrices, if :\mathbf = \begin 2 & 0 & 6 \\ 1 & -1 & 2 \end \,,\quad \mathbf = \begin 8 & -3 & 2 \\ 4 & 1 & -5 \end then :\begin\langle \mathbf ,\mathbf\rangle_\mathrm & = 2\cdot 8 + 0\cdot (-3) + 6\cdot 2 + 1\cdot 4 + (-1)\cdot 1 + 2\cdot(-5) \\ & = 21 \end


Complex-valued matrices

For two complex-valued matrices, if :\mathbf = \begin 1+i & -2i \\ 3 & -5 \end \,,\quad \mathbf = \begin -2 & 3i \\ 4-3i & 6 \end then :\begin \langle \mathbf ,\mathbf\rangle_\mathrm & = (1-i)\cdot (-2) + 2i\cdot 3i + 3\cdot (4-3i) + (-5)\cdot 6 \\ & = -26 -7i \end while :\begin \langle \mathbf ,\mathbf\rangle_\mathrm & = (-2)\cdot (1+i) + (-3i)\cdot (-2i) + (4+3i)\cdot 3 + 6 \cdot (-5) \\ & = -26 + 7i \end The Frobenius inner products of A with itself, and B with itself, are respectively :\langle \mathbf, \mathbf \rangle_\mathrm = 2 + 4 + 9 + 25 = 40 \qquad \langle \mathbf, \mathbf \rangle_\mathrm = 4 + 9 + 25 + 36 = 74


See also

*
Hadamard product (matrices) In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimension as the operand ...
*
Hilbert–Schmidt inner product In mathematics, Hilbert–Schmidt may refer to * a Hilbert–Schmidt operator; ** a Hilbert–Schmidt integral operator In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open ...
*
Kronecker product 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 generalization of the outer product (which is denoted by the same symbol) from vectors to ...
*
Matrix analysis In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix mu ...
*
Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
*
Matrix norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
*
Tensor product of Hilbert spaces In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly spea ...
– 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


References

{{DEFAULTSORT:Matrix Multiplication Matrix theory Bilinear maps Multiplication Numerical linear algebra