MA10210: ALGEBRA 1B
http://people.bath.ac.uk/aik22/ma10210
Comments on Sheet 9
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Revise eigenvectors corresponding to eigenvalues
with multiplicity > 1.
Be careful with calculation.
Work out what you need to find.
Try not to abandon answers halfway through.
Look at solutions to get an idea of how the
equations are set up to be more easily solved.
Demo of Q2 (a)?
Comments on Sheet 9
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Matrix multiplication doesn’t (always) obey the
cancellation law.
Some results would really make things easier, but
simply aren’t true.
A
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few of these appeared in determinant calculations...
Writing is an important aspect of mathematics
 even
if you chose maths to avoid essays...
Comments on Sheet 9
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Two-way implications
 You
need to show both ways,
 except in the case where every step is “if and only if”
(in which case why bother doing the second direction?)
Warm-up Question
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Q1:
 (i)
Find determinant of
, find eigenvalues.
 (ii) Use this to find the eigenvectors satisfying
.
 Use the eigenvectors to find P.
 (iii) Having diagonalised A, find An=PDnP.
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Q4:
 Note:
eigenvalues/eigenvectors satisfy Av=λv.
Warm-up Question
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Q1:
 Find
the characteristic polynomial and hence the
algebraic multiplicities.
 Think about the null space to calculate geometric
multiplicities.
 If
a 3x3 matrix has an eigenvalue with g.m. 3, what
must it be? (Think about the diaonalisation.)
 Calculate
the matrix, then its eigenvalues.
Warm-up Question
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Q3:
 Find
the eigenvalues of the two matrices.
 How many are positive? Negative?
 What does this represent?
Overview of Sheet 10
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Q2: similar to part of Q1. Don’t forget to say
whether the matrix is diagonalisable and why.
Q3: find a matrix A such that vn=Anv0.
Q4: see example in lecture notes.
Q5:
 (i)
What is matrix mult. (consider an individual term)?
 (ii) Use (i). Can be done (convincingly) in three steps.
 (iii) As well as stating how to define it, explain why the
definition makes sense.
Overview of Sheet 10
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Q6:
Note the minus sign! 
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Take the matrix given and calculate the
characteristic polynomial
 (ii) If the eigenvalues are as given, what must the
characteristic polynomial be?
 (i)