Wikipedia:Reference desk/Archives/Mathematics/2012 October 8

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October 8

How do you decompose this into orbits?

Let $M$ be the set of $m\times n$ matrices, and let $G=GL(\mathbb {R} ^{m})\times GL(\mathbb {R} ^{n})$, where an element $(P,Q)\in G$ acts on an element $A\in M$ by $(P,Q)A=PAQ^{-1}$. How do you decompose $M$ into $G$-orbits? I see no general way of doing it, since you can have general invertible P and Q, and then general A. I can only get special cases, for example that the zero matrix in $M$ is in an orbit by itself. Widener (talk) 13:48, 8 October 2012 (UTC)

The orbit of a matrix A consists of matrices that can be obtained from A by a sequence of elementary row and column operations. We can get a unique representative of each orbit as follows. Start with A. If there is a nonzero entry in the matrix, we can switch rows and columns to move this element to the upper left corner, and row multiplication to make this element 1. Then we use row and column addition to make all other entries in the first row and column zero. We repeat the same process with the $(m-1)\times (n-1)$ submatrix omitting the first row and column. In this way, we show that each orbit contains a matrix whose main diagonal starts with r 1s (where $r\leq \min\{m,n\}$), and the rest of the matrix is zero. This r is unique: the action of G preserves the rank of the matrix, hence r equals the rank of A. In summary, there are exactly $1+\min\{m,n\}$ orbits, and each orbit consists of matrices of rank r for some $r\leq \min\{m,n\}$. This works over any field, not just $\mathbb {R}$.—Emil J. 15:09, 8 October 2012 (UTC)

Thank sWidener (talk) 04:05, 9 October 2012 (UTC)

4 of a kind probabilities

In Texas Hold'em, what are the odds of hitting all 13 individual 4 of a kind hands? — Preceding unsigned comment added by 66.215.136.120 (talk) 17:15, 8 October 2012 (UTC)

Zero. You only get seven cards, so you can only hit at most one 4 of a kind hand. Of course that's not the answer you are looking for, but if you want a better answer, you'll have to ask the question more clearly. Looie496 (talk) 20:30, 8 October 2012 (UTC)

If you are asking about the odds of doing so in 13 out of 13 sequential hands, then we can do that math, given the probability P, of getting 4 of kind in each individual hand, and assuming you return the cards to the deck and reshuffle after each hand, with no jokers or other wilds:

Hand  Odds of getting 4 of a kind you haven't already had
----  ---------------------------------------------------
  1      P
  2      P(12/13)
  3      P(11/13)
  4      P(10/13)
  5      P(9/13)
  6      P(8/13)
  7      P(7/13)
  8      P(6/13)
  9      P(5/13)
 10      P(4/13)
 11      P(3/13)
 12      P(2/13)
 13      P(1/13)

This multiplies out as P¹³(12!/13¹²) = (0.00002056)P¹³. See Poker probability (Texas hold 'em) for some possible values to use for P (depends on several things). If we assume you don't put the cards back in the deck after each hand, then the odds go up (the very last hand is guaranteed to be a 4 of a kind in the one remaining rank). StuRat (talk) 20:50, 8 October 2012 (UTC)