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I have two matricies which represenet 2D convolutional filters, lets call them $W_1$ and $W_2$.

I want to make the inner product of the "flattened" versions of these variables to be 0. I.e. $flatten(W_1)^T flatten(W_2)=0$.

Is there a compact way to write this expression without using "flatten" funciton where the flatten function returns a columm vector of the matrix.

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The inner product is also defined for matrices and it’s basically the same.

$$\text{flatten}(A) \cdot \text{flatten}(B) = \langle \text{flatten}(A), \text{flatten}(B) \rangle = \langle A,B \rangle$$

Check out Frobenius Scalar Product

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Another common notation is using the trace operator, since $$ \langle A, B\rangle = {\rm flatten}(A)^T \cdot {\rm flatten}(B)= {\rm trace}\{A^T B\}$$ for any real-valued $A, B$.

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