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Based on the selection of questions set in past years, we should be able to do each of these problems set out below. Assume a non-singular matrix A has an eigenvector e and associated eigenvalue v.  Let D be a diagonal matrix. We have shown that A  = PDP-1 where P is constructed from the values of e and that the diagonal of D includes the various values of v.

1  Show A¹⁰= PD¹⁰ P^-1. Comment on Aⁿ. If necessary use the matrix A below.

2  State the eigenvectors and corresponding eigenvalues of kA, for real number k.

3  Let A be the 3x3 matrix with rows (5  4  1)  (-6  -2  3)  (8  8  3) and Q be the matrix with rows (1  -2  1) (2  -1  -1)  (-1  1  2), possibly as shown below:

           5    4    1                                         1     -2    1
A is   -6    -2    3                       Q is           2    -1   -1
           8    8    3                                         -1   1     2

a)  Convert A into a cubic equation.
b)  Given A  = PDP^-1, find 
c)  Matrix B is such that B = QAQ^-1. Find the eigenvalues and eigenvectors for B, where Q is shown above.

4  Given matrix A has eigenvectors e_i and eigenvalues v_i, and  B has e_i and w_i, then

a.  Show A+B has e_i and v_i + w_i

b.  Find eigenvectors and eigenvalues for aA + bB

c.  Find eigenvectors and eigenvalues for (aA + bB)⁴

d.  What about C = A + bI ?

e.  What about A + B² ? A + C² ?

5  Consider the inverse matrix A_-1, where A, e and v are defined as before
a.  Is e an eigenvector for A^-1? 
b.  Is v^-1 an eigenvalue of A^-1?
c.  Is v + v^-1 an eigenvalue for A + A^-1?
d.  What about (v + v^-1)³ ?

6  Given A  = PDP^-1, find E and Q such that  A + A² + A³ = Q^-1EQ. If appropriate for you, use the matrices A and P from Q3.

7  If C is the 3x3 matrix with rows (1  2  3) (0  2  7/2) (-3  6  0) then show that the latent roots (another term for eigenvalues) are integers a,b,c such that   a+2b+c=0 and find the matrices D and P so that C  = PDP^-1.

8  Does it matter whether you do post-multiplication,  (PQ)R or pre-multiplication, P(QR)?

Generally one is supposed to do pre-multiplication (“do this”, verb first). Assume D is a diagonal matrix, that A is a more general matrix and P is the matrix constructed from the eigenvectors of A. Consider the eight combinations   (PA)P^-1, P^-1(AP), P(AP^-1), (P^-1A)P, (PD)P^-1, (P^-1D)P, P(DP^-1), P^-1(DP). They fall into pairs and these pairs equate to A, to D or to Q or R (different, less obviously related matrices). Identify which pair equates to which result and try to explain why this is so. If necessary, use the matrix A from Q3. You may wish to prove your answer to the first line.

Answers? You jest....