[kubb]

> However, the multiplication of a cracovian by another cracovian is defined differently: the result of multiplying an element from column i of the left cracovian by an element from column j of the right cracovian is a term of the sum in column i and row j of the result.

Am I the only one for whom this crucial explanation didn’t click? Admittedly, I might be stupid.

Wikipedia is a bit more understandable: „The Cracovian product of two matrices, say A and B, is defined by A ∧ B = (B^T)A

[andrewla]

Agreed -- "is a term of the sum" is such an inverted way to look at it.

Better I think would be to say "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian".

And even by this definition the example given doesn't seem to track (and the strangeness of sometimes saying "+" and sometimes not, and having both "0" and "-0" in the example is bananas!):

   {  3  2 } {  1  -4 }  =   {  5   -2 }
   { -1  0 } { -2   3 }  =   {  0    2 }


   3 * 1 + -1 * -2 == 5 -- check
   3 * -4 + -1 * 3 == -15 -- what?
   2 * 1 + 0 * -2 == 2 (okay, but shouldn't this be in the lower left, column 1 dotted with column 2?)
   2 * -4 + 0 * 3 = -8 (now I'm really missing something)

[mci]

I took the liberty to replace my awkward wording with your "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian". Hope you don't mind. Thanks!

[mci]

Thanks for the feedback, everyone. I pasted my Polish text into Gemini to translate it into English. Gemini hallucinated the translation of this example. Now it should be OK.

[pomian]

Even in Polish, this comes out Greek to me.

[pomian]

I mean, it makes some sort of visual sense, but can't grasp the results from the matrices shown.

[AdamH12113]

The example is simply wrong, according to other sources. This along with the inconsistent formatting makes me wonder if it was written by an LLM. It's a shame; this seems like an interesting topic.

[tempodox]

You're not the only one. That “explanation” is just really bad.

[tgv]

It's the crucial part, and even with the example, I couldn't understand it. Like I can't understand why the second column in the first matrix doesn't have signs. Or why the 0 in the result matrix is negative.

But in another link I found that it's column by column multiplication. So A × B = C, then C[i][j] = sum(A[k][i] * B[k][j]). Unfortunately, the example doesn't match that definition...

[burnished]

No, I think it is too ambiguous to be useful. The example wasn't helpful either, I think they needed to perform the individual calculations for clarity.

[kubb]

Yeah, usually you name the matrix elements a, b, c, d, etc. and write out the formula for the elements of the result.