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Complex matrices

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jcanny edited this page Jan 29, 2013 · 19 revisions

Basics

BIDMat includes a complex, single precision matrix type called CMat. Real/complex pairs are stored in consecutive memory words as floats. There is however, no complex scalar type. Similar to Matlab, a complex scalar is realized as a 1x1 matrix.

To create a complex value, you can use the "i" method like this: 

scala> val b = 3 + 4.i
b: BIDMat.CMat =
  3+4i
which is evidently a CMat with a single value. Note that we used the "." operator to invoke the "i" method rather than using a space: 
scala> val b = 3 + 4 i
b: BIDMat.CMat =
  7i
since in Scala's syntax, "+" has a higher precedence than "i". 
scala> val b = 3 + (4 i)
b: BIDMat.CMat =
  3+4i
also works, as does (somewhat surprisingly) "3 + 4.0.i" However "3 + 4i" does not parse.

To create a large matrix of complex numbers, you can use any of the random number generators: 

scala> val c = rand(4,10) + rand(4,10).i
c: BIDMat.CMat =
   0.21497+0.77080i   0.67449+0.28513i   0.93723+0.71991i...
   0.37809+0.83195i  0.051680+0.89084i  0.038354+0.76330i...
   0.36922+0.82463i   0.78521+0.27711i   0.67254+0.71312i...
   0.87246+0.25732i   0.14137+0.19628i   0.98884+0.96585i...

Extracting Parts

To extract the real or imaginary parts of a complex matrix, use the operators ".r" and ".i". Like this: 

scala> val a = rand(4,10) + rand(4,10).i
a: BIDMat.CMat =
   0.67636+0.34410i   0.15675+0.46375i   0.43748+0.57167i...
   0.31279+0.63679i   0.69485+0.59093i   0.91233+0.15906i...
   0.10547+0.46058i   0.88596+0.73404i   0.58793+0.86155i...
   0.83065+0.87015i   0.84817+0.18380i  0.080891+0.78927i...

scala> a.r
res1: BIDMat.FMat =
   0.67636   0.15675   0.43748  0.081511   0.46293  0.097704   0.71535...
   0.31279   0.69485   0.91233   0.87120   0.12652   0.71330   0.35587...
   0.10547   0.88596   0.58793   0.90858   0.45308   0.45136  0.069531...
   0.83065   0.84817  0.080891  0.022294   0.73676   0.14168   0.91742...

scala> a.i
res2: BIDMat.FMat =
   0.34410   0.46375   0.57167   0.64560   0.92401   0.32618   0.44249...
   0.63679   0.59093   0.15906   0.68576   0.35331   0.12405   0.16163...
   0.46058   0.73404   0.86155   0.19624   0.87816   0.48652   0.56278...
   0.87015   0.18380   0.78927  0.065433   0.45076   0.87441   0.56792...

The operator ".i" therefore has two different meanings: When applied to an FMat, it returns a CMat which is sqrt(-1) times the original matrix. When applied to a CMat, it extracts the imaginary part and returns it as an FMat. This is done for simplicity but it could be confusing. If so, the operator ".im" is an alias which will extract the imaginary part of a CMat.

Operations

Most of the operations defined for real matrices will work on complex matrices (algebraic operators, concatenation, slicing, find functions). But comparison operators (<,>,==,>=,<=,!=) will not since complex numbers have no linear ordering.

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