University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B24S Fall 2011
Suppose that T : V −→ V is a linear transformation from vector space
V with ordered basis B into the vector space V with ordered basis B . Let ′
[T] B, B′ be the matrix representation of T relative to B amd B . Now let
′ ′ ′
T : V −→ W be a linear transformation from the vector space V to the
vector space W with ordered basis B . Then [T ◦T] ′ ′′= [T ]′ ′ ′[T] ′
B,B B , B B, B
This follows from the fact that for all v ∈ V and v ∈ V we have that
1. (T(v)) B′ = [T] B,B ′vB
′ ′ ′ ′
2. (T (v )) B” = [T ] B ,B′v B′
3. let v = T(v)
′ ′ ′ ′
4. (T ◦T)(v) B” = (T (T(v))) B” = [T ] B B” (T(v)) B′ = [T ] B ,B′′[T]B,B ′vB
Now suppose that B and B are two ordered bases of V and T : V −→ V
is a linear transformation.
Suppose that C B, B ′ is the change of coordinate matrix from B to B .
Then for all v ∈ V we have v ′ = C ′v . Suppose also that C ′ is
B B, B B B , B
the change of coordinate matrix from B to B. Then for all v ∈ V we have
v B C B , Bv B .
If [T] B is the matrix rep of T relative to B only i.e. T(v) B = [T] vB B
for all v ∈ V. and that [T] B ′ is the matrix rep of T relative to B only i.e.
T(v) B′ = [T] B ′vB ′ for all v ∈ V. Then we have that
[T]B= C B, B′[T]B ′C B, B′
Thus [T] ′ and [T] are similar matrices.
1 Remember that D is a similar matrix to A if D can be obtained from
A by performing elementary row (or column) operations. Now we see that
they are also both matrix rep of the same linear transformation T relative
to suitable diﬀerent ordered basis.
In the same way then it may be possible to ﬁnd the matrix rep of T rel
to some ordered basis such that this matrix is a diagonal.
Remember also that similar matrices have the same eigenvalues. (ie. λ is
an eigenvalue of A if Av = λv for some eigenvector v of A) We say that a
lin trans T has an eigenvalue λ if T(v) = λv and we call v an eigenvector of
Review Chapter 5 from last year.
It is sometimes of interest and convenient to have the matrix rep of a
linear transformation relative to an ordered basis B in a diagonal form. This
is only possible if a matrix rep of the linear transformation relative to a
known basis is diagonalizable.
Suppose we have A the matrix rep of T relative to some known basis
B . We need to ﬁnd if it is possible to ﬁnd a diagonal matrix which is the
representation of T relative to some basis B we need to ﬁnd out if A is
Thus we need to do the following:
1. We ﬁnd the characteristic polynomial |A − λI|.
2. We ﬁnd the eigevalues of A which are the roots of the characteristic
polynomial of A.
3. Find the algebraic multiplicity of each of the eigenvalues.
4. For each eigenvalue λ ﬁni the eigenspace E λiwhich is the nullspace of
A − λ .i
5. The dim(E ) iλithe geometric multiplicity of λ . i
6. If the geometric multiplicity of λ = tie algebraic multiplicity of λ for i
each λ ,ithen A is diagonalizable!
2 Once we determine that A is diagonalizable, then we construct C the
matrix whose columns are the bases vectors of each eigenspace E . Theλi
D = C AC is a diagonal matrix. The jth column of D has a nonzero entry
the eigenvalue whose eigenvector is the column j of the matrix C.
We have also established that A = [T] B′ and that D is a representation
of T relative to some basis B as well. In other words: D = C −1 [T] C ′
B, B′ B B, B
Thus C =