MA10210: ALGEBRA 1B

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Presentation transcript:

MA10210: ALGEBRA 1B

Comments on Sheet 9  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  Matrix multiplication doesn’t (always) obey the cancellation law.  Some results would really make things easier, but simply aren’t true.  A 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  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  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 A n =PD n P.  Q4:  Note: eigenvalues/eigenvectors satisfy Av= λ v.

Warm-up Question  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  Q3:  Find the eigenvalues of the two matrices.  How many are positive? Negative?  What does this represent?

Overview of Sheet 10  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 v n =A n v 0.  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  Q6:  (i) Note the minus sign!  Take the matrix given and calculate the characteristic polynomial  (ii) If the eigenvalues are as given, what must the characteristic polynomial be?