Frobenius Inner Products

01Combine matrix contributions into one number#

The Frobenius inner product multiplies matching entries of equal-shaped matrices and sums the products into one scalar. It is the matrix counterpart of a vector dot product.

The Hadamard product gives the array of individual products. Summing that array produces the inner product. Stopping before the sum returns the wrong kind of object.

02Include every position#

Let $A$ have rows $(1,-2),(0,3)$ and $B$ have rows $(4,1),(5,-1)$. The entrywise products are $4,-2,0,-3$. Their sum is $-1$.

Only summing the diagonal products would give $4-3=1$ and miss the off-diagonal contribution of $-2$. Every position belongs in the sum, including positions outside the diagonal of a rectangular matrix.

03A matrix-shaped weighted sum#

If $A$ contains sensitivities and $B$ contains small changes at matching positions, the inner product combines their signed contributions. For now, treat those matrices as supplied data; the calculus behind that interpretation has its own lesson.

You could list both matrices' entries in the same fixed order and take a vector dot product. The result would be identical. Mixing different entry orders changes the pairings and can change the scalar.

The inputs must have equal shapes. A negative inner product is valid: positive and negative contributions can outweigh one another. It is not a norm or an entrywise matrix output.