Any useful analog to finite differences for matrix products?

[Stephen Tashi]

Various closed form formulas for summing the first n terms of a sequence $\{a_i\}$ of numbers can be developed by considering the various order differences of the terms, such as ${\triangle} a_i = a_{i+1} - a_i$ and $\triangle^2 a_i = \triangle ( \triangle a_i)$. Closed form formulas occur if the m th and higher order differences of the terms vanish for some m.

If we have a sequence of invertible matrices $\{A_i \}$ can anything interesting be said about finding the product of the first n terms using information about the quotients of consecutive terms? ( $\delta A_i = A_{i+1} A_{i}^{-1}$ )?

 

[Stephen Tashi]

I'll try to clarify the question!

Each line in a different table of real numbers represents the the difference between the two adjacent entries in the line below it. If a difference table has an entire line of zeroes then each line has its own "closed form formula" for the sum of the first n terms in that line.

For example, the table whose last line of terms are defined by the formula $f(n) = n^2$ looks like

______0_____0_____0 ...
____2____ 2____ 2___...
___3__ _5____7 ___9 ...
1___ 4____ 9____ 16 ___25 ..

We know there are closed form formulas for $\sum_{i=1}^n n^2$ and $\sum_{i=1}^n (2n+1)$ etc.

I've never seen a similar theory for a "quotient table" of real numbers ( where each entry would be the quotient of the the two adjacent entries in the line below it)

For example:

___1____1_____1___1 ...
__3___3___3_____3______3...
7___21__63__189___567__1701_...

I suspect that there would be a closed form formula for the product of the first n terms in each line, but the theory of that formula would just amount to adding powers of various factors. That addition would fall under the scope of difference tables.

If we make tables whose entries are matrices instead of real numbers, a difference table would merely amount to a set of difference tables for the individual matrix entries.

However, a quotient table for matrices might be very interesting.

It would have the form

______1_______1_..
____A____A________A_...
__B___BA___BA^2_________BA^3
C__CB___CB^2A__CB^2ABA^2__CB^2ABA^2BA^3

So my question is whether any theory has been developed for the product of the first n terms in a line of such a table. Or is the theory of it just a trivial consequence of a powerful general theorem in linear algebra that I don't recognize?

 

[Studiot]

Are you trying to prove the multinomial theorem the hard way?

Difference equations lead to the use of continuous products in their solution

It also is possible to have "partial Difference equations" which are difference equations so that

We can define partial operators E and F

E_z (x,y) = z(x+1, y)

F_z (x,y) = z(x, y+1)

with difference operators

E = Δ_x and F = Δ_y

 

[Stephen Tashi]

Are you trying to prove the multinomial theorem the hard way?

What did you have in mind for a multinomial theorem about matrices?

 

[Studiot]

What did you have in mind for a multinomial theorem about matrices?

I'm not sure about matrices but the multinomial theorem includes such products.


$${({x_1} + {x_2} + {x_3} + ...{x_m})^n} = \sum {\frac{{n!}}{{{n_1}!{n_2}!...{n_m}!}}} x_1^{{m_1}}x_2^{{m_2}}...x_m^{{n_m}}$$


I was rather taken with your idea.

 

[Stephen Tashi]

Perhaps the entries in a "quotient table" of real numbers have a relation to that multinomial theorem. However, since matrix multiplication is, in general, non-commutative, I don't think that theorem applies to products of matrices because a term such as $A^2 B A$ can't be "combined" with a term like $A^3 B$. Maybe there is a generalization of the multinomial theorem.