Re: squareroots of matrices

Probably best to start with

$\displaystyle \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \times \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}=\begin{pmatrix} p & q \\ r & s \\ \end{pmatrix}$

You can get 4 simultaneous equations by multiplying out the matrices. And a 5th equation $\displaystyle p+s \leq x$.

Your simultaneous equations will be quadratic so when finding solutions you will get square roots of numbers, you can find more inequalities by ensuring that the square roots are the square root of a positive number.

You might also be able to get equations by knowing that all elements are natural numbers. For example, if you found that $\displaystyle p+r=1+\frac{d}{b+c}$ since p+r must be a natural number d must be a multiple of b+c, so you can say that d=k(b+c)

Re: squareroots of matrices

Quote:

Originally Posted by

**Shakarri** Probably best to start with

$\displaystyle \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \times \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}=\begin{pmatrix} p & q \\ r & s \\ \end{pmatrix}$

You can get 4 simultaneous equations by multiplying out the matrices. And a 5th equation $\displaystyle p+s \leq x$.

You mean like:

p = aa + bc

r = ac + dc

q = ab + ab

s = bc + dd

?

and p + s < x because of the trace

Quote:

Your simultaneous equations will be quadratic so when finding solutions you will get square roots of numbers, you can find more inequalities by ensuring that the square roots are the square root of a positive number.

Sry don't get your clue there

Quote:

You might also be able to get equations by knowing that all elements are natural numbers. For example, if you found that $\displaystyle p+r=1+\frac{d}{b+c}$ since p+r must be a natural number d must be a multiple of b+c, so you can say that d=k(b+c)

[/QUOTE]

Ah ok, I understand the schematic here but dont really get how that helps to determine the quantity of the matrices..

Thank you very much for answering, btw I'm not necessarily ambitious in developing a own solution, so if somebody could give me a detailed step by step solution without executing every single calculation , would be great. Just want to see how its done correctly.

Thanks in advance for your help!

Re: squareroots of matrices