In mathematics, the Frobenius inner product is a binary operation takes two matrices and returns a number. It is often denoted ⟨ A , B ⟩ F . The operation is a component-wise inner product of two matrices as though they are vectors. 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
A = ( A 11 A 12 ⋯ A 1 m A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n m ) , B = ( B 11 B 12 ⋯ B 1 m B 21 B 22 ⋯ B 2 m ⋮ ⋮ ⋱ ⋮ B n 1 B n 2 ⋯ B n m ) the Frobenius inner product is defined by the following summation Σ of matrix elements,
⟨ A , B ⟩ F = ∑ i , j A i j ¯ B i j , = t r ( A T ¯ B ) where the overline denotes the complex conjugate. Explicitly this sum is
⟨ A , B ⟩ F = A ¯ 11 B 11 + A ¯ 12 B 12 + ⋯ A ¯ 1 m B 1 m + A ¯ 21 B 21 + A ¯ 22 B 22 + ⋯ + A ¯ 2 m B 2 m + A ¯ n 1 B n 1 + A ¯ n 2 B n 2 + ⋯ + A ¯ n m B n m The calculation is very similar to the dot product, which in turn is an example of an inner product.
It is a sesquilinear form, for four complex-valued matrices A, B, C, D, and two complex numbers a and b:
⟨ a A , b B ⟩ 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 . For two real-valued matrices, if
A = ( 2 0 6 1 − 1 2 ) , B = ( 8 − 3 2 4 1 − 5 ) then
⟨ A , B ⟩ F = 2 ⋅ 8 + 0 ⋅ ( − 3 ) + 6 ⋅ 2 + 1 ⋅ 4 + ( − 1 ) ⋅ 1 + 2 ⋅ ( − 5 ) = 16 + 12 + 4 − 1 − 10 = 21 For two complex-valued matrices, if
A = ( 1 + i − 2 i 3 − 5 ) , B = ( − 2 3 i 4 − 3 i 6 ) then the complex conjugates (without transpose) are
A ¯ = ( 1 − i + 2 i 3 − 5 ) , B ¯ = ( − 2 − 3 i 4 + 3 i 6 ) and
⟨ A , B ⟩ F = ( 1 − i ) ⋅ ( − 2 ) + ( + 2 i ) ⋅ 3 i + 3 ⋅ ( 4 − 3 i ) + ( − 5 ) ⋅ 6 = ( − 2 + 2 i ) + − 6 + 12 − 9 i + − 30 = − 26 − 7 i while
⟨ B , A ⟩ F = ( − 2 ) ⋅ ( 1 + i ) + ( − 3 i ) ⋅ ( − 2 i ) + ( 4 + 3 i ) ⋅ 3 + 6 ⋅ ( − 5 ) = − 26 + 7 i 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 The inner product induces the Frobenius norm
∥ A ∥ F = ⟨ A , A ⟩ F . 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 (denoted by "vec", converted into column vectors) as follows,
v e c ( A ) = ( A 11 A 12 ⋮ A 21 A 22 ⋮ A n m ) , v e c ( B ) = ( B 11 B 12 ⋮ B 21 B 22 ⋮ B n m ) , the matrix product
v e c ( A ) T v e c ( B ) = ( A ¯ 11 A ¯ 12 ⋯ A ¯ 21 A ¯ 22 ⋯ A ¯ n m ) ( B 11 B 12 ⋮ B 21 B 22 ⋮ B n m ) reproduces the definition, therefore
⟨ A , B ⟩ F = v e c ( A ) T v e c ( B ) . 