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.


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


[citation needed]


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]



Frobenius norm[edit]

The inner product induces the Frobenius norm



Real-valued matrices[edit]

For two real-valued matrices, if


Complex-valued matrices[edit]

For two complex-valued matrices, if



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

See also[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.