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Concept
Frobenius inner product
Summary
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. Explicitly this sum is
The calculation is very similar to the dot product, which in turn is an example of an inner product.
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.
Official source
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