Question

Good methods for finding eigen values?

\(M =
\begin{array}{cc}
0&1&1&1&1& \\
1&0&1&1&1 \\
1&1&0&1&1 \\
1&1&1&0&1\\
1&1&1&1&0
\end{array}\)

Answer 1

@freckles where you go ;(

Answer 2

\[\det(A- \lambda I) =0\] and then use cofactor expansion like crazy

Answer 3

I said good methods :)

Answer 4

=(

Answer 5

row operations first? idk that may destroy the determinant...

Answer 6

lol
well ... I was kind of trying to see if could find a pattern
but I didn't see a pattern
like you know M_n is a n by n matrix with 0's on the diagonal and 1's everywhere else
but it was totally not a good method

Answer 7

and not just a bad method but it didn't seem to work like that

Answer 8

does this whole matrix break one of the rules for row operations? Don't we need a 1 at the top left corner of the first row?

Answer 9

word

Answer 10

row swaps... so it does not break the first rule

Answer 11

I would switch the last one with the first one r_1<->r_5

Answer 12

swap r_2 <-> r_4 , then r_3 <-> r_4 ?

Answer 13

actually I do see a pattern

Answer 14

so there must be some kinda of shortcut for M_n where M_n is a n by n matrix with 0's a long the diagonal and 1's elsewhere

Answer 15

I want to make a conjecture
\[\text{ egienvalues }(M_n)=n-1,-1 \text{ (multiplicity } (n-1) \text{ )}\]

Answer 16

Now I know this isn't a method
but it might help to come up with something from this
if it is even true

Answer 17

like we know the trace of M is suppose to be the sum of the eigenvalues

Answer 18

\[0=\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5\]

Answer 19

I don't know a quick way to find the determinant
but the product of the eigenvalues equals the determinant

Answer 20

ill play with this too

Answer 21

another cute thing
\[M_n=[1]_n-I\]
where 1_n is a n by n matrix only made of 1's
and I is the identity matrix
don't know if this helps yet

Answer 22

you know I'm not totally sure yet but it seems to be:
in this case anyways
\[\text{ eigenvalues(}M_n\text{)}=\text{ egivenvalues(}[1]_n\text{)}-\text{ egienvalues(}I\text{)}\]

Answer 23

the eigenvalues of I are 1 (n amount of times)

and

the eigenvalues of [1]_n are n, 0 (n-1 amount of times)