How come mathematicians will make up some stupid bullshit so that you can take the square root of a negative number, but I can't multiply a 3x3 matrix by a 4x4 matrix? I don't get it.
>>17050551sure you can, just extend all finite matrices to infinite matrices with 1's down the diagonal and 0's padding out the rest and you can multiply matrices of any size to your heart's content. It's even still associative.
>>17050551>it just is okay?
>>170505514 derivations away from mass gap
>>17050551You can. Just define a new generalized matric product.
>>17050551>I can't multiply a 3x3 matrix by a 4x4 matrix?okay how would you do it>>17050553so what size is the product of 3x3 * 4x4?>>17050568okay, how?
>>17050601>so what size is the product of 3x3 * 4x4?Most natural choice would be 3x4
because you're using the canonical definition of matrix multiplication. You could make up some stupid bullshit to do that, no one will probably care though.
>>17050553>>17050568>You can, just make it upand math is supposed to be the "universal truth". LOOOOOOOOOOOOOOL
>>17050551https://en.wikipedia.org/wiki/Kronecker_product>>17050601>so what size is the product of 3x3 * 4x4?12 * 12[eqn]\mathbf{A}\otimes\mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B}\end{bmatrix} = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\ a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq}\end{bmatrix}. [/eqn]
>>17050627that's the stupidest equation I've ever seen, wtf is the point of that?
>>17050551wouldn't you multiply it by a negative number or it's quantum crossing over with the positive. That is another meaning.
>>17050551AI will fix it.
>>17050629>wtf is the point of that?the point was to answer OP's question
>>17050627nice
>>17050629It's how matrix operations behave in spaces of uncorrelated variables. For example, a vector space of functions of 2 coordinates can often be expressed in a basis of functions of the form h(x,y)=f(x)g(y). Then something like d/dx * d/dy (both expressed as matrices in that space) will behave like the equation he wrote.
>>17050551>make up some stupid bullshit so that you can take the square root of a negative numberThat's actually a side effect of trying to make a number system larger than the reals. It's the only possible solution that doesn't introduce infinite hierarchies (hyperreals and the like).
