Linear Algebra Eigenvalue problem

[hushed ruin]

hello! i'm having a bit of trouble on part b of this linear algebra problem regarding eigenvalues in overlapping eigenspaces. basically i'm confused on where to start. i understand that the vector exists but i'm lost on how we compute AB

[thin galleon]

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[hushed ruin]

my work so far has been trying to logic my way through a calculation. Av = λa • v and Bv = λb • v

im stuck on what to do from here though. i understand i have to find the vector, but im worried im over complicating the problem

[olive widget]

You want to show $(AB)v=(\lambda_A\lambda_B)v$

[cosmic basin]

Omegabet_

[hushed ruin]

yes

thats what i'm confused about

[olive widget]

so compute (AB)v

Hint: matrix * is associative

[hushed ruin]

unfortunately i'm confused on how to compute AB, should i define 2 matrices, and compute that? then i wont know how to show that λa•λb is also ther

[olive widget]

apply associativity to (AB)v, what do you get?

[hushed ruin]

well obviously we get that (λaλb)v, but i dont see why we can just do that

[olive widget]

No

do what I said

what does associative mean?

[hushed ruin]

(a•b)•c = a•(b•c)

[olive widget]

so do that here

what is (AB)v equal to?

[hushed ruin]

OHHHHHHH

[olive widget]

yeah

[hushed ruin]

A(Bv)

etc

[olive widget]

and Bv is?

[hushed ruin]

λbv

then repeat

[olive widget]

yes

QED

[hushed ruin]

thank you i'm so sorry 😭

+close

[teal sonnet]

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[teal sonnet]

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