# Eigenvalue and diagonalisation question

1. Apr 3, 2012

### spaghetti3451

1. The problem statement, all variables and given/known data

Find the eigenvalues and eigenvectors of the following matrix: M =
1 1
0 1
Can this matrix be diagonalised?

2. Relevant equations

3. The attempt at a solution

The characteristic equation is $(1 - \lambda)^{2} = 0$ which gives $\lambda = 1$. Substitute $\lambda = 1$ and eigenvector = {x,y} into the eigenvalue equation gives the two equations x+y = x and y = y. The first equation implies that y = 0. The second equation is redundant. So, x is free to assume any complex value. So, the eigenvalue is 1 and the eigenvector is {1,0}.

I think everything I have done so far is fine. If it isn't, please point out.

The problem starts with the second part: 'Can this matrix be diagonalised?' I know that to diagonalise a matrix is equivalent to changing the basis of the matrix and the eigenvectors. I am not quite sure I get this or the fact that the eigenvectors have to span the ??? to accomplish the diagonalisation.

Thanks in advance for any help.

2. Apr 3, 2012

### tiny-tim

hi failexam!
changing the basis doesn't change the behaviour …

the space spanned by the eigenvectors of the original matrix is one-dimensional (the line y = o)

how many dimensions is the space spanned by the eigenvectors of a diagonal matrix?

(alternatively, if it was diagonalised, what would those diagonal entries be?)

3. Apr 3, 2012

### HallsofIvy

Staff Emeritus
An n by n matrix is "diagonalizable" if and only if it has n independent eigenvectors. If a matrix is NOT diagonalizable, it can be put in "Jordan Normal Form". And the matrix in this problem already is in Jordan Normal Form!