Question

We are told that A = ( 7 2 -4 1 ) has eigenvalues 5 and 3 with respective eigenvectors ( 1 -1 ) and ( 1 -2 ) . So we can diagonalise A by finding an invertible matrix M and diagonal matrix D such that M^-1 AM = D . i) If we wanted D to be the matrix D = ( 5 0 0 3 ) then we should take M = <<1|1>,<-1|-2>> ii) If instead we wanted D to be the diagonal matrix D = ( 3 0 0 5 ) then we should take M = <<1|1>,<-2|-1>> iii) One of the advantages of having a diagonal matrix is that matrix powers are easy to take, for example ( 5 0 0 3 )^6 = <<15625|0>,<0|729>> The relation for A^n is not nearly as simple. iv) Once we write A = MDM^-1 we observe that A^2 = MD^2M^-1, A^3 = MD^3M^-1, and so on. Thus, using parts (i) and (iii), we can easily calculate that the top left entry of A^6 is Recall: the Maple notation for the matrix ( 1 2 3 4 ) is <<1 | 2 >, <3 | 4 >>.

          We are told that A = ( 7 2 
 -4 1 ) has eigenvalues 5 and 3 with respective eigenvectors ( 1 
 -1 ) and ( 1 
 -2 ) . So we can diagonalise A by finding an invertible matrix M and diagonal matrix D such that
M^-1 AM = D .
i) If we wanted D to be the matrix D = ( 5 0 
 0 3 ) then we should take
M = <<1|1>,<-1|-2>>
ii) If instead we wanted D to be the diagonal matrix D = ( 3 0 
 0 5 ) then we should take
M = <<1|1>,<-2|-1>>
iii) One of the advantages of having a diagonal matrix is that matrix powers are easy to take, for example
( 5 0 
 0 3 )^6 = <<15625|0>,<0|729>>
The relation for A^n is not nearly as simple.
iv) Once we write A = MDM^-1 we observe that A^2 = MD^2M^-1, A^3 = MD^3M^-1, and so on. Thus, using parts (i) and (iii), we can easily calculate that the top left entry of A^6 is 
Recall: the Maple notation for the matrix ( 1 2 
 3 4 ) is <<1 | 2 >, <3 | 4 >>.

We are told that A = ( 7 2
-4 1 ) has eigenvalues 5 and 3 with respective eigenvectors ( 1
-1 ) and ( 1
-2 ) . So we can diagonalise A by finding an invertible matrix M and diagonal matrix D such that
M^-1 AM = D .
i) If we wanted D to be the matrix D = ( 5 0
0 3 ) then we should take
M = <<1|1>,<-1|-2>>
ii) If instead we wanted D to be the diagonal matrix D = ( 3 0
0 5 ) then we should take
M = <<1|1>,<-2|-1>>
iii) One of the advantages of having a diagonal matrix is that matrix powers are easy to take, for example
( 5 0
0 3 )^6 = <<15625|0>,<0|729>>
The relation for A^n is not nearly as simple.
iv) Once we write A = MDM^-1 we observe that A^2 = MD^2M^-1, A^3 = MD^3M^-1, and so on. Thus, using parts (i) and (iii), we can easily calculate that the top left entry of A^6 is
Recall: the Maple notation for the matrix ( 1 2
3 4 ) is <<1 | 2 >, <3 | 4 >>.

Added by Susan W.

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