Interesting Matrix Identity

In summary, the matrix S_{ij} = 2^{-(2N - i - j + 1)} \frac{(2N - i - j)!}{(N-i)!(N-j)!} contains a numerically unstable maximum when N=1...15.
  • #1
madness
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Hi all,

I've come across an interesting matrix identity in my work. I'll define the NxN matrix as [tex] S_{ij} = 2^{-(2N - i - j + 1)} \frac{(2N - i - j)!}{(N-i)!(N-j)!}.[/tex] I find numerically that [tex] \sum_{i,j=1}^N S^{-1}_{ij} = 2N, [/tex] (the sum is over the elements of the matrix inverse). In fact, I expected to get 2N based on the problem I'm studying, but I don't know what this complicated matrix expression is doing or why it equals 2N. Does any of this look familiar to anyone here?

Thanks for your help!

P.S. If this is in the wrong subforum, please move it.
 
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  • #2
Interesting, how did you come across this? using some numerical computing software like matlab?

@fresh_42 or @Mark44 might be interested in how you discovered this.
 
  • #3
I haven't run through the math, but keep in mind that the inverse matrix element can be expressed as:
$$(S^{-1})_{ij} = \frac{1}{\det{S}}C_{ji}$$
where ##C_{ji}## is the element of the transposed cofactor matrix. Also remember that the determinant can be expressed as a cofactor expansion:
$$\det{S} = \sum_{i=1}^{N} S_{ij} C_{ij}$$
Also keep in mind that the cofactor expansion works for any row or any column, so that
$$(S^{-1})_{ij} = \frac{C_{ji}}{ \sum_{i=1}^{N} S_{ji} C_{ji}}$$
I dunno, maybe that helps. It might not hurt, too, to see if you can pull out a general formula for the cofactor.
 
  • #4
Thanks for the help.

@jedishrfu I discovered this trying to maximise the following:

[tex] \frac{\left[ \int_0^\infty f(t) dt \right]^2}{\int_0^\infty f^2(t) dt } [/tex] where [tex] f(t) = \sum_{i=1}^N w_i \frac{(ct)^{N-i}}{(N-i)!} e^{\lambda t} [/tex] and [tex] w_i [/tex] are weights which I want to maximise with respect to. I can show that the maximum is [tex] \frac{1}{-\lambda} \sum_{ij} \left(S^{-1}\right)_{ij} [/tex] and using Matlab this turns out to be [tex]\frac{2N}{-\lambda}[/tex] for N=1...15 (I stopped here as it became numerically unstable).

@TeethWhitener I can see that your approach must give the right answer, but finding a closed form expression for the cofactor seems difficult for an arbitrary NxN matrix.
 

What is the "Interesting Matrix Identity"?

The "Interesting Matrix Identity" is a mathematical equation that relates the determinants of two matrices to the determinants of their product. It is also known as the "Matrix Determinant Identity" or the "Matrix Multiplication Identity".

Who discovered the "Interesting Matrix Identity"?

The "Interesting Matrix Identity" was first discovered by the German mathematician Carl Friedrich Gauss in the early 1800s.

What is the significance of the "Interesting Matrix Identity"?

The "Interesting Matrix Identity" is significant because it allows for the simplification of calculations involving determinants of matrices. It also has important applications in linear algebra and is used in the proof of other mathematical theorems.

Can the "Interesting Matrix Identity" be applied to matrices of any size?

Yes, the "Interesting Matrix Identity" can be applied to matrices of any size as long as the matrices are square and have the same dimensions.

Are there any limitations or exceptions to the "Interesting Matrix Identity"?

Yes, the "Interesting Matrix Identity" only applies to square matrices and cannot be used for non-square matrices. Additionally, the matrices involved must have the same dimensions in order for the identity to hold true.

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