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> 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 · 8h ago

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 · 7h ago

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 · 8h ago

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 · 7h ago

Even in Polish, this comes out Greek to me.

pomian · 6h ago

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

AdamH12113 · 8h ago

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 · 8h ago

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

tgv · 8h ago

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 · 8h ago

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 · 8h ago

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

fxj · 8h ago

I didnt get the explanation of the multiplication. After reading the wikipedia article it made mode sense:

https://en.wikipedia.org/wiki/Cracovian

The Cracovian product of two matrices, say A and B, is defined by

A ∧ B = BT A,

where BT and A are assumed compatible for the common (Cayley) type of matrix multiplication and BT is the transpose of B.

Since (AB)T = BT AT, the products (A ∧ B) ∧ C and A ∧ (B ∧ C) will generally be different; thus, Cracovian multiplication is non-associative.

A good reference how to use them and why they are useful is here (pdf):

https://archive.computerhistory.org/resources/access/text/20...

adastra22 · 7h ago

As far as I can tell I don’t think it is correct to say that this isn’t a matrix. B is just written down in transposed form. Whether that makes the math more or less clear is something you can argue for or against, but it’s the same math and it is confusing to call it something else.

noosphr · 4h ago

It is a tensor of rank two with a special binary operation on tensors. These objects aren't matrices in the mathematical sense any more than convolution kernels aren't.

adastra22 · 46m ago

A tensor of rank two is the same thing as a matrix…

noosphr · 26m ago

It isn't.

Matrices come with the matrix product defined over them.

This is one of four possible closed first order tensor tractions for an order two tensor, viz. AijBik AijBki AijBjk AijBkj. Only the third is applicable to matrices, all other contractions only work for general tensors without transposition.

What we deal with in computer science are actually n dimensional arrays since we don't have the co and contravariant indices that define tensors in physics.

gnulinux · 8h ago

I guess I'm skeptical of using a non-associative algebra instead of something that can trivially be made into a ring or field (i.e. matrix algebra). What advantages does this give us?

mci · 8h ago

Author here. There are no practical advantages, as far as I know. Not even faster multiplication on today's computers.

hansvm · 6h ago

One thing that comes up in the sort of code ML I like to write is a careful attention to memory layout. Cracovians, defined according to some sibling comment as (B^T)A, make that a little more natural, since B and A can now have the same layout. I haven't used them though, so I don't have a good sense of whether that's more or less painful than other approaches.

bravesoul2 · 5h ago

Shouldn't be the same on a computer right? The change is in human perception not actually what hapens when multiplying.

noosphr · 4h ago

In Einstein notation this operation is Aij Bkj, which incidentally shows why Einstein notation is so useful.

Syzygies · 3h ago

I'm a mathematician who taught linear algebra for decades. I love Einstein notation. I don't find Cracovians interesting at all.

Old texts got really worked up whether a vector was a row or column. The programming language APL resolved this quite nicely: A scalar has no dimensions, a vector has its length as its one dimension, ... Arbitrary rank objects all played nicely with each other, in this system.

A Cracovian is a character or two's difference in APL code. There's a benign form of mental illness learning anything, where one clutches onto something novel and obsesses over it, rather than asking "That was exciting! What novel idea will I learn in the next five minutes?" I have friends from my working class high school who still say "ASSUME makes an ask of you and me" as if they just heard it for the first time, while the most successful mathematicians that I know keep moving like sharks.

I wouldn't stall too long thinking about Cracovians, as amusing a skim as the post provided.

noosphr · 3h ago

I mean theres nothing special about naming binary operations on tensors of fixed rank. Matrices have some nice mathematical properties which is why they are studied so much in mathematics. But for number crunching there is no reason to prefer then to cracovians, or vice versa, without knowing what the underlying memory layout is in hardware.

TimorousBestie · 7h ago

What an interesting little nook of matrix analysis history! Thanks for sharing. :)

esafak · 7h ago

Missed a chance to call it the twisted sister!

semiinfinitely · 4h ago

Uhh so it's just matrices where the left slot of matmul is transposed?