How to find eigenvalues and eigenvectors without using built-in functions (Ex. eig)?

Let's assume I have a 2x2 matrix defined by:

A = [ 7 3; 3 -1 ];

How can I find the eigenvectors and eigenvalues of matrix A without using built-in functions? More specifically, without using eig()?

[James Tursa]

What techniques are you supposed to use? Please post the complete wording of the homework assignment so we know what you are allowed to use.

[Gri Sch]

I should have clarified in the initial post, I'm only allowed to use arithmetic operators. The eigenvalues and eigenvectors are a small portion of the assignment, and the only thing I'm having trouble with.

A reworded ask would be: How can I find the eigenvalues and eigenvectors of a 2x2 matrix by only using arithmetic operators?

I'm very new to MatLab so my understanding of what is/isn't allowed by the program is limited.

[James Tursa]

Are you supposed to find eigenvalues by doing root finding on the characteristic equation? Are you supposed to be finding eivenvectors using the power method? There has to be more to your instructions that you can tell us.

[Gri Sch]

I have to perform a coloring transformation on a 1000x2 matrix. A step in this transformation is to obtain the eigenvectors and eigenvalues from a given covariance matrix. Given that my only task is to perform the coloring transformation, the method in which I obtain the eigenvectors and eigenvalues is not specified and does not matter, as long as I only use arithmetic operations. The assignment document actually doesn't even mention eigenvectors or eigenvalues anywhere, it is just a step in the coloring transformation.

Answer 1

[Steven Lord]

Without using any built-in functions? It depends -- do you consider the commands used in MEX-files to be built-in functions? In computing the eigenvalues and eigenvectors from scratch you're likely going to need to do arithmetic and all the operators (+ or plus, - or minus, * or mtimes, etc.) are built-in functions.

Specifically without using eig? Since I'm guessing this is part of a homework assignment your textbook likely include pseudocode for an algorithm to find eigenvalues and eigenvectors. If you encounter difficulty while implementing that pseudocode, show us the code you've written so far and describe specifically where you're having difficulty and what difficulty you're having and we may be able to offer suggestions.