Can someone explain why λ₁ is -2 and not -1?

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[kindred totem]

$$ = (1+\lambda) (2+\lambda)(3-\lambda) = 0 $$

[long spindle]

aL

[kindred totem]

so lambda =-1,-2 or 3

@cerulean sonnet

[cerulean sonnet]

https://youtu.be/H74flsYlQ1w?si=CRA4huht4q3Qa4nD the guy in this video got λ₁ = -2, how is it -2 instead of -1?

[kindred totem]

why does it have to be -1

that's the end result he arrives at

he shows the initial matrix is similar to D, i.e diagonalized with respect to the calculated basis

[cerulean sonnet]

This is how I did it

[kindred totem]

that's not correct

[cerulean sonnet]

What did I do wrong?

[kindred totem]

you have a factor 1- lambda

there is no such factor

[cerulean sonnet]

Okay I’ve put the minus sign

Before -lambda

[kindred totem]

show me

[kindred totem]

now you have your eigenvalues

[cerulean sonnet]

So should λ₁ not be -1 instead of -2?

[kindred totem]

why

why not 3 even?

[cerulean sonnet]

So that it’s in order?

[kindred totem]

are you concerned about the order of the factors?

[cerulean sonnet]

Yes

[kindred totem]

multiplication is commutative

[cerulean sonnet]

The order doesn’t matter?

[kindred totem]

try it

pick them in different order on the diagonal and follow the algorithm

you will get a different P

but it's nonetheless similar to the initial matrix

[cerulean sonnet]

Will the final answer be the same?

[kindred totem]

no

[cerulean sonnet]

As that of the video?

[kindred totem]

.

[cerulean sonnet]

So lambda 1 HAS to be -2?

[kindred totem]

no

[kindred totem]

.

[cerulean sonnet]

I’m so lost

[kindred totem]

he just picked this order

but it's not mandatory

the end goal is to diagonalize the given matrix

[cerulean sonnet]

Why though? That’s what I’m trying to understand

[kindred totem]

that means by definition you have to show it's similar to a diagonal matrix

[cerulean sonnet]

Just because?

[cerulean sonnet]

Alright

[kindred totem]

pick D = (3,-2,-1) if you want

you still get the same eigenspaces for each eigenvalue of course, but when you construct P, the column vectors are swapped around