Eigenvalues: find the eigenvalues of the n-by-n matrix....

[nandizzle]

Wassup,

We're doing eigenvalues in class atm, and though I can find them for finite matrices, I'm struggling with ones which extend to the nth row:

e.g:

Find the eigenvalues of the nxn matrix:

1 1 ... 1
1 1 ... 1
...
...
...
1 1 ... 1

Any help at all is appreciated :wink:

 

[galactus]

I believe this is just a matter of recognition.

Your characteristic polynomial will be \(\displaystyle \L\\(-1)^{n}({\lambda}^{n}-n{\lambda}^{n-1})\)

For instance, if n=5, \(\displaystyle \L\\5{\lambda}^{4}-{\lambda}^{5}\)

If n=4, \(\displaystyle \L\\{\lambda}^{4}-4{\lambda}^{3}\)

See the pattern?.

The eigenvalues easily follow.

 

[Vampire99]

legend

 

[nandizzle]

hmmm, I'm not sure if this is what you mean, but does it then easilty follow the eigen values are '0' and 'n'

 

[galactus]

Yep