Frobenius inner product

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In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted . The operation is a component-wise inner product 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[edit]

Given two complex number-valued n×m matrices A and B, written explicitly as

the Frobenius inner product is defined as,

where the overline denotes the complex conjugate, and denotes Hermitian conjugate.[1] Explicitly this sum is

The calculation is very similar to the dot product, which in turn is an example of an inner product.[citation needed]

Relation to other products[edit]

If A and B are each real-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 ""), then

Therefore

[citation needed]

Properties[edit]

It is a sesquilinear form, for four complex-valued matrices A, B, C, D, and two complex numbers a and b:

Also, exchanging the matrices amounts to complex conjugation:

For the same matrix,

,[citation needed]

and,

.

Frobenius norm[edit]

The inner product induces the Frobenius norm

[1]

Examples[edit]

Real-valued matrices[edit]

For two real-valued matrices, if

then

Complex-valued matrices[edit]

For two complex-valued matrices, if

then

while

The Frobenius inner products of A with itself, and B with itself, are respectively

See also[edit]

References[edit]

  1. ^ a b Horn, R.A.; C.R., Johnson (1985). Topics in Matrix Analysis (2nd ed.). Cambridge: Cambridge University Press. p. 321. ISBN 978-0-521-83940-2.