Linear Algebra question

Fuzzywhale

For my PhD research I have come up with a giant system of quadratic forms that I would like to solve.

I have absolutely no idea how to tackle this . So any help math people could provide would be wonderful!

Editing for more info:

So this is some tensor algebra; I am unraveling an indexed equation into all of its components. I have maybe 30 equations that are of the form

x_1x_2+x_3_x4+x_5x_6+x_7x_8+X_9X_10+X_11X_12+X_13_X14+X_15X_16+X_17X_18+X_19X_20+X_21X_22-X_23X_24=0

Does Mathematica/Matlab/Maple Have any magic that can handle this? Alternatively can anyone recommend a nice linear algebra book? Granted I've only sat through 2 linear algebra courses but this sort of thing never came up for me.

lifeincognito

While my math is probably not nearly up to snuff for this, you might want to talk to people who know about optimization of quadratic forms? You'll have to set some sort of bounds or conditions on these functions if you hope to solve them because each equation has two unknowns. So unless these equations are inter-related, which I think you are implying from the Linear Algebra title, you'll have a rough go at it. Otherwise if they ARE related you can just start performing basic math ( addition, subtraction, etc) to attempt to work them down into an equation with only one unknown. However, it seems like someone pursing a PhD would already have tried that so I am confused. Honestly you'll probably have to give a bit more information to illicit strong posts from the forum.

A brief google search turned up this fine document from MIT. Which may not be helpful as it is mostly math jargon, but you must know some math if you PhD requires you to deal with quadratic forms. It seems to deal with solving matrices of quadratic forms so maybe it'll help? Either way you get a bump from my post. Good luck with your research.

Fuzzywhale

thanks very much for this link im checking it out right now.

Yeah I'm going to hit up some department members on monday. This has been my "thing to do" for maybe 3 weeks and i've kept botching it. I'm pretty clueless

CrystalMethodist

Can't you just put those equations into a matrix and solve them by Gaussian elimination?

If you have 24 variables, you should be able to specify coefficients on all of them and then add everything together. If you don't have that many unique products, you can make the products into new variables (like a variable called X_1*X_2) and then solve again once you have constraints on the products.

Fuzzywhale

That was of course my first though as well but:
Isn't that only valid for linear equations, that is, something like y=aX_1+bX_2+......?


Edit: Duh I see what you mean. I will try some substitutions.