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MATB24H3 Lecture Notes - Diagonal Matrix, Diagonalizable Matrix, Linear Map


Department
Mathematics
Course Code
MATB24H3
Professor
Sophie Chrysostomou

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University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B24S Fall 2011
Lecture 9
Suppose that T:VVis a linear transformation from vector space
Vwith ordered basis Binto the vector space Vwith ordered basis B. Let
[T]B, Bbe the matrix representation of Trelative to Bamd B. Now let
T:VWbe a linear transformation from the vector space Vto the
vector space Wwith ordered basis B′′ .Then [TT]B,B′′ = [T]B, B′′ [T]B, B
This follows from the fact that for all vVand vVwe have that
1. (T(v))B= [T]B,BvB
2. (T(v))B= [T]B,B′′ v
B
3. let v=T(v)
4. (TT)(v)B= (T(T(v)))B= [T]BB(T(v))B= [T]B,B′′ [T]B,BvB
Now suppose that Band Bare two ordered bases of Vand T:VV
is a linear transformation.
Suppose that CB, Bis the change of coordinate matrix from Bto B.
Then for all vVwe have vB=CB, BvB. Suppose also that CB, B is
the change of coordinate matrix from Bto B. Then for all vVwe have
vB=CB, BvB.
If [T]Bis the matrix rep of Trelative to Bonly i.e. T(v)B= [T]BvB
for all vV. and that [T]Bis the matrix rep of Trelative to Bonly i.e.
T(v)B= [T]BvBfor all vV. Then we have that
[T]B=C1
B, B[T]BCB, B
Thus [T]Band [T]Bare similar matrices.
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Remember that Dis a similar matrix to Aif Dcan be obtained from
Aby performing elementary row (or column) operations. Now we see that
they are also both matrix rep of the same linear transformation Trelative
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. λis
an eigenvalue of Aif Av=λvfor some eigenvector vof A) We say that a
lin trans Thas an eigenvalue λif T(v) = λvand we call van eigenvector of
T.
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 Bin 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 Athe matrix rep of Trelative to some known basis
B. We need to find if it is possible to find a diagonal matrix which is the
representation of Trelative to some basis Bwe need to find out if Ais
diagonalizable.
Thus we need to do the following:
1. We find the characteristic polynomial |AλI|.
2. We find the eigevalues of Awhich are the roots of the characteristic
polynomial of A.
3. Find the algebraic multiplicity of each of the eigenvalues.
4. For each eigenvalue λifind the eigenspace Eλiwhich is the nullspace of
Aλi.
5. The dim(Eλi) is the geometric multiplicity of λi.
6. If the geometric multiplicity of λi= the algebraic multiplicity of λifor
each λi, then A is diagonalizable!
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