Question

How do I compute a determinant using Leibniz's (big and inefficient) formula?

Answer 1

I'm talking about this ( http://en.wikipedia.org/wiki/Leibniz_for … ) but, the Wikipedia link is too "proofy" and, there is no regular hands-on problem so, I'm having trouble getting it.

Any input would be greatly appreciated!

Answer 2

Leibniezs formula eh

what do you know of it so far?

Answer 3

http://answers.yahoo.com/question/index?qid=20130325071913AApgrux

seems to have been another resource you used? maybe?

Answer 4

Yes, there isn't much useful (for me) stuff that I can find about this.

All I know is that it involves products, I think, (n-1)! times and that it works only for square matrices.

All I think I need is super trivial examples with 2x2 and maybe 3x3 matrices or whatever you think is best.

Answer 5

so the formula looks a little something like this correct?
\[\Large det(A)=\sum_{\sigma \in S_n} sgn(\sigma)\prod_{i=1}^{n}A_{i,\sigma_i}\]

Answer 6

It seems to be correct but, I'm not sure. Wikipedia has the stuff inside/on the right of the product symbol as a_sigma(i), i.

While doing a trivial example, could you please explain what the notation is doing?

Answer 7

from what i can tell, there are permutation that can be had to account for sigma. but i got no idea where they are getting it from.

they say of n=3, then sigma = [1,2,3], or [3,2,1], or [1,3,2] etc ... and they assign an even or odd to a permutation based on some unknown concept.

the sign of the sigma is either a + or a - depending on if its an even or odd parity maybe?

the product is then just the matrix element of the ith row and the column of the ith sigma.

i assume from this that the n is the number corresponding to an nxn matrix

say a 2x2 is:\[\Large det(A)=\sum_{\sigma \in S_n} sgn([1,2])\prod_{i=1}^{2}A_{i,\sigma_i}\]
\[\Large det(A)=sgn([1,2])(A_{1,1}*A_{2,2})+sgn([2,1])(A_{1,2}*A_{2,1})\]

Answer 8

Sorry for the potentially stupid question but, what is sgn([1,2])?

Answer 9

You want the determinant correct?

Answer 10

Yes (using this technique - not any technique).

Answer 11

im not so sure what the innards are, they have to do with permuations dealing with an entry inside of the matrix; and the sgn indicates "sign" as either negative of postive.

Answer 12

i think it equates to the standard:\[(-1)^{i+j}\text{ for a given} ~a_{i,j}~\text{entry in A}\]

Answer 13

if i+j is even, sgn is +
if i+j is odd, sgn is -

this conforms the subdeterminants into proper values without haveing to adjust the i,j to 1,1 each time

Answer 14

Is the sigma a subscript to the A as well?

Answer 15

\[A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\]
subdeterminant for a1,1 is +d
subdeterminant for a1,2 is -c
subdeterminant for a2,1 is -b
subdeterminant for a2,2 is +a

Answer 16

the sigma notation is shorthand since the subdeterminant of nxn matrix can grow rather large

Answer 17

(I was asking because the subscript has its own subscript.)

Also, sorry if I ask dumber questions than usual but, my brain is worn out from thinking too much.:
I didn't understand what you said here:
"subdeterminant for a1,1 is +d
subdeterminant for a1,2 is -c
subdeterminant for a2,1 is -b
subdeterminant for a2,2 is +a"

Answer 18

the determinant of a matrix is composed of subdeterminants .... according to some rule that i cant quite recall

Answer 19

i remember ....

Answer 20

take the value at Ai,j, and multiply it by its subdeterminant value

then add up all the Ai,j subdeterminant values ... perhaps

Answer 21

Use matrix([1,2],[3,4]) as an example.

Answer 22

please

Answer 23

\[A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\]

A1,1 = 1

is subdeterminant is the value of the determinant obtained from corssing out the row and column that contain A1,1; in this case we are left with the determinant of |4| = 4, then move across the columns or rows taking a sub determinant along the way

Answer 24

so det(A) = 1|4| - 2|3| - 3|2| + 4|1| is what i read the setup as, but thats 2 times as much

Answer 25

Don't take this in a bad way but, I need to go now. I'll be back as soon as I can to read what you said. (Don't wait for me unless you'll be here anyways.)