Some problem on eigenvalues,eigenvectors and diagonalizability

I'm preparing final exam on linear algebra and come across with these problem:

#1

Let A be an n*n matrix.Assume that for some , and .

a)Find all the eigenvalues of A.

b)Prove that A is not diagonalizable.

c)Prove that k<n

d)If k=n-1,find the dimension of null A

#2

Let A be an n*n matrix and be two real numbers.Assume that

Prove that A is diagonalizable.

#3

Let

and

be eigenvectors correspond to distinct eigenvalues

of an n*n matrix A.

Prove that

is not an eigenvector of A.

For #3,

Suppose on the contrary,

is an eigenvector of A

Then there exists a number n such that

Since eigenvectors are lin. ind.

therefore,

Contradict to

are distinct.

Is it right ?

Re: Some problem on eigenvalues,eigenvectors and diagonalizability

Quote:

Originally Posted by

**maoro** Let A be an n*n matrix.Assume that for some

,

and

.

a)Find all the eigenvalues of A.

If is an eigenvalue of there exists such that . Then, which implies

Quote:

b)Prove that A is not diagonalizable.

etc ...

Try the rest of problem #1

P.S. Please, see rule #8 here:

http://www.mathhelpforum.com/math-he...hp?do=vsarules

Re: Some problem on eigenvalues,eigenvectors and diagonalizability

Re: Some problem on eigenvalues,eigenvectors and diagonalizability

Sorry for posting too many questions on a single thread.

btw, I have not covered Jordan canonical forms and so is there any alternative approach for #2 ?

Re: Some problem on eigenvalues,eigenvectors and diagonalizability