with
non zero entries only on the leading diagonal, it is very easy to
find powers of
If
for example,
then
A Level Maths Notes: FP4 – Powers of Matrices
Given a matrixwith
non zero entries only on the leading diagonal, it is very easy to
find powers ofIf
for example,then
A matrix
is
said to be diagonalizable if we can find a matrix
such
that
or
equivalently
where
is
a diagonal matrix with non zero entries only on the leading diagonal.
Of course only square matrices can be diagonalized, but a wide range
of square matrices can be diagonalized. If a matrix is
diagonalizable, then
since when the brackets are removed each occurrence
of
reduces
to the identity matrix.
The matrixexists
in the case when the
timesmatrixhasindependent
eigenvectors. The eigenvectors form a linearly independent set, so a
matrixwhose
columns consist of them can be inverted thenfound
using
Example: The matrix
has
eigenvalues 1 and 3 with eigenvectors
and
so
that
and
D is the matrix with diagonal entries equal to the
eigenvalues, with the eigenvalues in the column of that corresponding
eigenvector:
is
then given by
