Eigenvalue and diagonalisation question

[spaghetti3451]

Homework Statement

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

Homework Equations

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.

 

[tiny-tim]

hi failexam!

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.

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?)

 

[HallsofIvy]

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!