Discussion about eigen vectors and eigen values

[gleaming panther]

I have doubts regarding the formula A-λI=0.
Like how does the formula aid in finding the eigen value. Not to look at the definition though. But how did we arrive to the conclusion that the above described formula would give us the value by which a given vector scales? On what grounds or process did we go through to realise the formula?

[carmine burrow]

This is a very used concept, so you should be able to find some resource online that explains it very well 🤔

Anyway, in short

We start with

$\mathbf{Ax} = \lambda \mathbf x$

[violet perch]

ELeonardo

[carmine burrow]

Then put everything to the left side

$\mathbf{Ax}-\lambda\mathbf x = 0$

[violet perch]

ELeonardo

[carmine burrow]

Now we factor x, remember Ix = x

$\left(\mathbf{A}-\lambda\mathbf I \right)\mathbf x= 0$

[violet perch]

ELeonardo

[carmine burrow]

There's a trivial solution for this, that is x = 0. This is the only solution for matrices that have full rank, or matrices that are invertible

These properties are linked to det(matrix) ≠ 0

We don't want the trivial solution, so we need det(...) = 0