Systems of linear equations

[glad lily]

Hi! Is there a fast way to solve systems of linear equations when matrix has complex numbers (e.g. 2+i)? I understand that 2x2 and 3x3 matrices can be solved by Cramer's rule, but it already requires time to multiply complex numbers. How to quickly solve systems with 3+ equations with 3+ variables with complex multipliers? Or basically how to quickly find a deteminant of 3x3 and higher dimension square matrices? (I've just started my first year in university so I dont know advanced linear algebra and advanced techniques)

Also if the method is Gaussian or Gauss-Jordan then how to quickly work with complex multipliers? Unless I am wrong, when complex numbers are in matrix then it is block matrix so all transformations of multipliers are transformation with matrices inside the main matrix

Please correct me if I am wrong, I haven't spent a lot of time on linear algebra yet so I can misunderstand some conceptions

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[vivid wedge]

Hm...
Well, I'd just use Gauss, both for linear systems and determinants.

[glad lily]

Could you suggest me sources where I can learn about Gauss for matrices that contain complex numbers? I had to solve 2x2 system and I tried to solve it with Gauss and I got confused during matrix transformation (when I solved default equations where all multipliers are real, I didn’t have any problems with Gauss (tho I haven’t dealt with general case with NxN system yet))

[vivid wedge]

I don't really know of any resources with that, but it's just the same principle - perform equivalent operations on rows until you bring the matrix to RREF (or REF, at least).

[glad lily]

Should I learn how to deal with block matrices since I have complex numbers? Because operations with such matrices confuse me (at least at this moment)

[vivid wedge]

Haven't had much experience with block matrices, but there are some helpful formulas for them.
In my case, I handle determinants larger than 3⨯3 using a computer 😅

I mean, I know how to compute determinants of any size, but it's just too time-consuming.

[glad lily]

Understandable :D I just understand that I might have a task to solve some crazy system with complex numbers so I want to get ready for it. I understand how to solve determinants when matrices have real components but complex components are unexplored field for me at this moment

[vivid wedge]

Just curious, is there some aspect you want to apply this to, or is this purely of theoretical interest?

[glad lily]

One of the reasons is that want to make my life easier when I will have to deal with more complex tasks. We have just started analytical geometry course (first year of bachelor) (first topic was complex numbers) and were given a 2x2 system of equations. When lecturer solved it, he had to go through a lot of transformations to get an answer while I did it quite quickly with Cramer's rule. Tho the thing that I am too lazy to calculate 3 or more multiplications of (x + iy) made me curious if there is a way to solve these things faster and then it has also gotten some patterns of curiosity to understand the general principle

[vivid wedge]

Oh, I see.
Well, Cramer's is fine for small systems, but bad for larger ones.

Though, just curious: what calculations are you doing in analytic geometry that involve complex matrices?

I've only used real ones.

[glad lily]

basically none, I would say these are my worms xD
We had a system of equations where all multipliers are complex numbers and for that moment we 'shouldn't' know anything about matrices. I just wanted to prepare for future linear algebra that will be in our second semester and start solving systems by using matrices and their properties

[vivid wedge]

You have analytic geometry before linear algebra? Interesting, it's usually either together or the other way around.

[glad lily]

as I understood, our program made analytic geometry be analytic geometry + an introduction to linear algebra, not sure tho. In first semester we have calculus (here we call it mathematical analysis) and analytic geometry

[vivid wedge]

Ah, ok. That seems like the usual, more or less.

Or, at least, similar to what I had.

[glad lily]

I don't know why but when I try to solve even 2x2 matrix with complex numbers by using Gauss, I cant get rid rather of real part or of imaginary part of a component so I just come just to another view of system which still has 2 variables in each equation (the only difference is that some multipliers have only real/imaginary part)

[vivid wedge]

That's because you need to use complex numbers in row operations, too.

So, if, say, we have a matrix of a system with a(11) = 1 and a(21) = 3 + 4i, then the first action would be II - (3 + 4i)*I.

[glad lily]

what is 'I' here?

also when I tried to do another thing, I got a triangle matrix which is solvable but it has already got several places where I will have to multiply 2 complex numbers (I gues the ammount of places where I will have to multiply 2 complex numbers will just grow higher and higher when I get to higher dimensions)

[vivid wedge]

I is first row, II is second row.

[vivid wedge]

Can you show what you mean?