Frobenius inner product

Short description: Binary operation, takes two matrices and returns a scalar

In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted ⟨𝐀,𝐁⟩F. 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.

## Contents

* 1 Definition
* 1.1 Relation to other products
* 1.2 Properties
* 1.3 Frobenius norm
* 2 Examples
* 2.1 Real-valued matrices
* 2.2 Complex-valued matrices
* 3 See also
* 4 References

## Definition

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

𝐀=(A11A12⋯A1mA21A22⋯A2m⋮⋮⋱⋮An1An2⋯Anm),𝐁=(B11B12⋯B1mB21B22⋯B2m⋮⋮⋱⋮Bn1Bn2⋯Bnm)

the Frobenius inner product is defined as,

⟨𝐀,𝐁⟩F=∑i,jAij‾Bij=Tr(𝐀T‾𝐁)≡Tr(𝐀†𝐁)

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

⟨𝐀,𝐁⟩F=A‾11B11+A‾12B12+⋯+A‾1mB1m+A‾21B21+A‾22B22+⋯+A‾2mB2m⋮+A‾n1Bn1+A‾n2Bn2+⋯+A‾nmBnm

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

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 "vec(⋅)"), then

vec(𝐀)=(A11A12⋮A21A22⋮Anm),vec(𝐁)=(B11B12⋮B21B22⋮Bnm),vec(𝐀)‾Tvec(𝐁)=(A‾11A‾12⋯A‾21A‾22⋯A‾nm)(B11B12⋮B21B22⋮Bnm)

Therefore

⟨𝐀,𝐁⟩F=vec(𝐀)‾Tvec(𝐁).[citation needed]

### Properties

Like any inner product, it is a sesquilinear form, for four complex-valued matrices A, B, C, D, and two complex numbers a and b:

⟨a𝐀,b𝐁⟩F=a‾b⟨𝐀,𝐁⟩F
⟨𝐀+𝐂,𝐁+𝐃⟩F=⟨𝐀,𝐁⟩F+⟨𝐀,𝐃⟩F+⟨𝐂,𝐁⟩F+⟨𝐂,𝐃⟩F

Also, exchanging the matrices amounts to complex conjugation:

⟨𝐁,𝐀⟩F=⟨𝐀,𝐁⟩F‾

For the same matrix,

⟨𝐀,𝐀⟩F≥0,[citation needed]

and,

⟨𝐀,𝐀⟩F=0⟺𝐀=𝟎.

### Frobenius norm

The inner product induces the Frobenius norm

‖𝐀‖F=⟨𝐀,𝐀⟩F.[1]

## Examples

### Real-valued matrices

For two real-valued matrices, if

𝐀=(2061−12),𝐁=(8−3241−5)

then

⟨𝐀,𝐁⟩F=2⋅8+0⋅(−3)+6⋅2+1⋅4+(−1)⋅1+2⋅(−5)=21

### Complex-valued matrices

For two complex-valued matrices, if

𝐀=(1+i−2i3−5),𝐁=(−23i4−3i6)

then

⟨𝐀,𝐁⟩F=(1−i)⋅(−2)+2i⋅3i+3⋅(4−3i)+(−5)⋅6=−26−7i

while

⟨𝐁,𝐀⟩F=(−2)⋅(1+i)+(−3i)⋅(−2i)+(4+3i)⋅3+6⋅(−5)=−26+7i

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

⟨𝐀,𝐀⟩F=2+4+9+25=40⟨𝐁,𝐁⟩F=4+9+25+36=74

## See also

* 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

## References

1. ↑ 1.0 1.1 Horn, R.A.; C.R., Johnson (1985) (in en). Topics in Matrix Analysis (2nd ed.). Cambridge: Cambridge University Press. pp. 321. ISBN 978-0-521-83940-2.

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