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DeterminantalRepresentations :: randomOrthogonal

randomOrthogonal -- constructs a random special orthogonal matrix

Synopsis

Description

This method returns a random orthogonal matrix of a given size $n$. The orthogonal matrix is constructed via Cayley's correspondence, which gives a bijection between skew-symmetric matrices, and orthogonal matrices $O$ which do not have $1$ as an eigenvalue (i.e., $O - I$ is invertible). Up to changing signs of rows, any orthogonal matrix can be obtained this way: if $G\cong (\ZZ/2\ZZ)^n$ is the group of diagonal matrices with diagonal entries equal to ±1, acting on $n\times n$ matrices by left multiplication, then (as one may check) every $G$-orbit contains a matrix that does not have $1$ as an eigenvalue (if the characteristic is not 2).

Note that the matrices which feature in the Cayley correspondence have determinant $(-1)^n$, so this method scales by $-1$ to return a special orthogonal matrix. Thus the matrices returned by this method do not have $-1$ as an eigenvalue.

By default a matrix over RR is returned. This method also accepts a ring as an (optional) argument, in which case a special orthogonal matrix over the ring is returned, with entries in the base coefficient ring.

i1 : O1 = randomOrthogonal 5

o1 = | .442386  -.0306554 .751799   -.317548 .370563  |
     | .133864  .768167   -.0944111 -.513838 -.345046 |
     | -.60855  -.385718  .234815   -.587034 -.284854 |
     | -.42001  .39448    .591227   .519553  -.220209 |
     | -.489525 .323399   -.145593  -.143489 .783576  |

                5          5
o1 : Matrix RR    <--- RR
              53         53
i2 : isOrthogonal O1

o2 = true
i3 : eigenvalues O1

o3 = {.070857+.997486*ii}
     {.070857-.997486*ii}
     {.803392+.595451*ii}
     {.803392-.595451*ii}
     {1                 }

o3 : VerticalList
i4 : det O1

o4 = 1

o4 : RR (of precision 53)
i5 : R = QQ[x,y]

o5 = R

o5 : PolynomialRing
i6 : O2 = randomOrthogonal(5, R)

o6 = | -22142569999/27593776175 90555408/788393605   -5532008344/27593776175
     | 193172688/788393605      145827307/157678721  24042978/788393605     
     | 591854008/5518755235     -5780466/157678721   -5151709442/5518755235 
     | -598559432/27593776175   -200993706/788393605 7495643483/27593776175 
     | -84039992/157678721      40244400/157678721   18443792/157678721     
     ------------------------------------------------------------------------
     -5025793592/27593776175  2046480376/3941968025 |
     -216954846/788393605     70436016/788393605    |
     -1194249041/5518755235   -206975072/788393605  |
     -25053509506/27593776175 -751964432/3941968025 |
     22251808/157678721       -123874657/157678721  |

             5       5
o6 : Matrix R  <--- R
i7 : isOrthogonal O2

o7 = true
i8 : det(O2), det(O2+id_(R^5))

          5225472
o8 = (1, ---------)
         157678721

o8 : Sequence

See also

Ways to use randomOrthogonal :

For the programmer

The object randomOrthogonal is a method function.