Question #23912
1 Answer
See below
Explanation:
To diagonalise, we need the matrix's eigenvectors. And .... the matrix needs to offer up 2 distinct eigenvectors in order to be diagonalisable.
Handy trick: For a triangular matrix, the eigenvalues are its diagonal entries.
But we can show that here by going the long way round: they are also the solutions to the characteristic equation, which for a
The eigenvectors (
# lambda = 1#
From the first equation,
# lambda = 4#
From the first equation,
The diagonalisation matrix P is then simply:
The matrix D is a diagonal matrix that has the eigenvalues as it only non-zero entries:
We are asserting that:
Now, we have everything apart from
So our assertion here is that:
Try it and see.
You can then do some seriously cool stuff with matrix. There is a point to diagonalisation :)