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{SECT 0 {EXCHG {PARA 240 "" 0 "" {TEXT 264 44 "Topics in Mathematical \+
Biophysics, Fall 2005" }{TEXT 264 0 "" }{TEXT 264 51 "\nMaple tutorial
 on eigenvalues of rotation matrices" }{TEXT 264 0 "" }}}{EXCHG {PARA 
241 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}
{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 20 "with(LinearAlgebra):" }
{MPLTEXT 1 265 0 "" }}}{SECT 1 {PARA 242 "" 0 "" {TEXT 266 18 "Eigenva
lues review" }{TEXT 266 0 "" }}{EXCHG {PARA 243 "" 0 "" {TEXT 267 188 
"In this section you will review what you learned about finding eigenv
alues in Maple.  Also you will learn how to pick out individual eigenv
alues and eigenvectors. You will also review the " }{TEXT 268 6 "evalf
 " }{TEXT 267 88 "command which gives you your answer in the exact num
ber of decimal places that you want." }{TEXT 267 0 "" }}}{EXCHG {PARA 
241 "> " 0 "" {MPLTEXT 1 265 33 "A := <<2,3,3>|<3,2,-5.>|<1,2,0>>;" }
{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 269 422 "The Eig
envector command gives a sequence of two things, the eigenvalues as a \+
vector and the corresponding eigenvectors as a matrix.  To get just th
e first term in the sequence,  the eigenvalues,  append a [1] to tthe \+
command.  This gives you just the eigenvalues.  To get just the second
 term in the sequence,  the eigenvectors,  append a [2] to the command
.  This gives you just the eigenvectors as columns of a matrix.. " }
{TEXT 269 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 267 55 "We find the e
igenvalues and eigenvectors of A. Use the " }{TEXT 268 3 "map" }{TEXT 
267 9 " and the " }{TEXT 268 5 "evalf" }{TEXT 267 93 " command (floati
ng point evaluate) to get the number of decimal places you want, say 4
.  The " }{TEXT 268 3 "map" }{TEXT 267 77 " command is used to apply a
 function to all entries of a vector or a matrix. " }{TEXT 267 0 "" }}
}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 104 "eigvals := map(x -> ev
alf(x,4),Eigenvectors(A)[1]); eigvects := map(x -> evalf(x,4),Eigenvec
tors(A)[2]);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 
270 42 "For example, to get the second eigenvalue." }{TEXT 270 0 "" }}
}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 22 "eigval2 := eigvals[2];"
 }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 194 "To ge
t a particular eigenvector from the matrix of eigenvectors, use the co
lumn command.  Here we ask for the third column, which is the third ei
genvector, corresponding to the third eigenvalue." }{TEXT 270 0 "" }}}
{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 31 "eigvect2 := Column(eigve
cts,2);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 267 
102 "The conjugate transpose of a matrix A, A*,  is sometimes called t
he adjoint.  In Maple the command is " }{TEXT 271 19 "HermitianTranspo
se." }{TEXT 267 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 272 29 "
HermitianTranspose(eigvect2);" }{MPLTEXT 1 272 0 "" }}}{EXCHG {PARA 
243 "" 0 "" {TEXT 267 4 "The " }{TEXT 268 14 "Linear Algebra" }{TEXT 
267 94 " package has a command to consturct a diagonal matrix out of t
he entries in a column vector.  " }{TEXT 267 0 "" }}}{EXCHG {PARA 241 
"> " 0 "" {MPLTEXT 1 265 33 "diag := DiagonalMatrix(<1,2,-3>);" }
{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 269 59 "The norm
 of a matix is largest absolute value of any entry." }{TEXT 269 0 "" }
}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 272 11 "Norm(diag);" }
{MPLTEXT 1 272 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 269 151 "A neat \+
way to show that a matris is nearly zero is to show that its norm is n
early zero.  If you want to show that A=B, show that A-B is nearly zer
o.  " }{TEXT 269 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 20 "Now tr
y problem 1.  " }{TEXT 270 0 "" }}}}{SECT 1 {PARA 242 "" 0 "" {TEXT 
266 15 "Complex numbers" }{TEXT 266 0 "" }}{EXCHG {PARA 243 "" 0 "" 
{TEXT 267 178 "Notice that the eigenvalues and eigenvectors of the abo
ve matrix are complex valued. Maple has commands to work with complex \+
numbers.  In Maple, a complex number is of the form  " }{TEXT 268 10 "
a + b*I.  " }{TEXT 267 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 68 "
The command abs gives the absolute value of a real or complex number" 
}{TEXT 270 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 272 13 "abs(2
 - 3*I);" }{MPLTEXT 1 272 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 267 
50 "To find the conjugate of a complex number use the " }{TEXT 268 10 
"conjugate " }{TEXT 267 8 "command." }{TEXT 267 0 "" }}}{EXCHG {PARA 
241 "> " 0 "" {MPLTEXT 1 265 19 "conjugate(2 - 3*I);" }{MPLTEXT 1 265 
0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 42 "You can also get real a
nd imaginary parts." }{TEXT 270 0 "" }}}{EXCHG {PARA 241 "> " 0 "" 
{MPLTEXT 1 265 24 "Re(2+ 3*I); Im(2 + 3*I);" }{MPLTEXT 1 265 0 "" }}}
{EXCHG {PARA 243 "" 0 "" {TEXT 267 43 "Get the conjugate of a vector b
y using the " }{TEXT 268 3 "map" }{TEXT 267 69 " command.  This takes \+
the conjugate of each coordinate of the vector." }{TEXT 267 0 "" }}}
{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 34 "map(conjugate, <2 + 3*I,
4 + 5*I>);" }{MPLTEXT 1 265 0 "" }}}}{SECT 1 {PARA 247 "" 0 "" {TEXT 
-1 14 "Random numbers" }{TEXT -1 0 "" }}{EXCHG {PARA 248 "" 0 "" 
{TEXT 244 12 "The command " }{TEXT 271 7 "rand() " }{TEXT 244 47 "retu
rns a random 12 digit non-negative integer." }{TEXT 200 0 "" }}}
{EXCHG {PARA 249 "> " 0 "" {MPLTEXT 1 0 7 "rand();" }{MPLTEXT 1 0 0 ""
 }}}{EXCHG {PARA 249 "> " 0 "" {MPLTEXT 1 0 13 "sin(rand()); " }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 250 "" 0 "" {TEXT 244 56 "to convert
 the sin of an integer to a decimal, use evalf" }{TEXT 244 0 "" }}}
{EXCHG {PARA 251 "> " 0 "" {MPLTEXT 1 0 6 "evalf(" }{MPLTEXT 1 -1 13 "
sin(rand()));" }{MPLTEXT 1 -1 0 "" }}}}{SECT 1 {PARA 242 "" 0 "" 
{TEXT 266 12 "Euler angles" }{TEXT 266 0 "" }}{EXCHG {PARA 243 "" 0 ""
 {TEXT 267 78 "First we construct functions giving rotation matrices f
or rotations about the " }{TEXT 273 1 "y" }{TEXT 267 5 " and " }{TEXT 
273 1 "z" }{TEXT 267 7 " axes. " }{TEXT 267 0 "" }}}{EXCHG {PARA 241 "
> " 0 "" {MPLTEXT 1 265 15 "Ry := theta -> " }{MPLTEXT 1 265 0 "" }
{MPLTEXT 1 265 64 "\n<<cos(theta),0,-sin(theta)>|<0,1,0>|<sin(theta),0
,cos(theta)>>:" }{MPLTEXT 1 265 0 "" }{MPLTEXT 1 265 16 "\nRz := theta
 -> " }{MPLTEXT 1 265 0 "" }{MPLTEXT 1 265 66 "\n<<cos(theta),sin(thet
a),0>|<-sin(theta), cos(theta), 0>|<0,0,1>>:" }{MPLTEXT 1 265 0 "" }}}
{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 12 "Ry(t),Rz(t);" }{MPLTEXT 
1 265 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 214 "You can get othe
r rotation matrices by multiplying together combinations of the above \+
three.    The following matrix gives rotation 5.2 radians about the z \+
axis followed by a rotations 2.1 radians about the x axis." }{TEXT 
270 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 265 62 "Euler := (al
pha, beta, gamma) -> Rz(alpha).Ry(beta).Rz(gamma);" }{MPLTEXT 1 265 0 
"" }}}{EXCHG {PARA 249 "> " 0 "" {MPLTEXT 1 0 26 "Euler(alpha, beta, g
amma);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 270 31 "N
ow try problems 2 and 3 and 4." }{TEXT 270 0 "" }}}}{EXCHG {PARA 241 "
> " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 252 "" 0 "" {TEXT 225 0 "" }}
{PARA 253 "" 0 "" {TEXT 210 0 "" }}{PARA 254 "" 0 "" {TEXT 274 0 "" }}
{PARA 255 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }