matrix square

see:  http://www.math.utexas.edu/pipermail/maxima/2012/031320.html
Square roots  of matrix

 Example1. Find square roots of matrix([33,24],[48,57])
(%i1) X:matrix([a,b],[c,d])$
(%i2) A:matrix([33,24],[48,57])$
(%i3) X.X-A;
(%o3) matrix([b*c+a^2-33,b*d+a*b-24],[c*d+a*c-48,d^2+b*c-57])
(%i4) append(%[1],%[2]);
(%o4) [b*c+a^2-33,b*d+a*b-24,c*d+a*c-48,d^2+b*c-57]
(%i5) solve(%);
(%o5)
[[d=-5,c=-8,b=-4,a=-1],[d=7,c=4,b=2,a=5],[d=5,c=8,b=4,a=1],[d=-7,c=-4,b=-2,a=-5]]
(%i6) makelist(ev(X,%[k]),k,1,4);
(%o6)
[matrix([-1,-4],[-8,-5]),matrix([5,2],[4,7]),matrix([1,4],[8,5]),matrix([-5,-2],[-4,-7])]
 Test:
(%i7) makelist(%[k]^^2-A,k,1,4);
(%o7)
[matrix([0,0],[0,0]),matrix([0,0],[0,0]),matrix([0,0],[0,0]),matrix([0,0],[0,0])]

 Example2. Find square roots of matrix([0.8,0.5],[0.2,0.5]);
 This problem also has four solutions.
(%i8) X:matrix([a,b],[c,d])$
(%i9) A:matrix([0.8,0.5],[0.2,0.5]);
(%o9) matrix([0.8,0.5],[0.2,0.5])
(%i10) ratprint:false$ algebraic:true$
(%i12) load(diag)$
(%i13) mat_function(sqrt,A);
(%o13)
matrix([(2*sqrt(3))/(7*sqrt(10))+5/7,5/7-(5*sqrt(3))/(7*sqrt(10))],[2/7-(2*sqrt(3))/(7*sqrt(10)),(5*sqrt(3))/(7*sqrt(10))+2/7])
(%i14) sr1:expand(rootscontract(ratsimp(%))); float(%);
(%o14)
matrix([sqrt(30)/35+5/7,5/7-sqrt(30)/14],[2/7-sqrt(30)/35,sqrt(30)/14+2/7])
(%o15)
matrix([0.8707778735729,0.32305531606774],[0.1292221264271,0.67694468393226])
 Other solution is:
(%i16)
sr2:matrix([5/7-sqrt(30)/35,sqrt(30)/14+5/7],[sqrt(30)/35+2/7,2/7-sqrt(30)/14]);
float(%);
(%o16)
matrix([5/7-sqrt(30)/35,sqrt(30)/14+5/7],[sqrt(30)/35+2/7,2/7-sqrt(30)/14])
(%o17)
matrix([0.55779355499852,1.10551611250369],[0.44220644500148,-0.10551611250369])
(%i18) sr3:-sr1$
(%i19) sr4:-sr2$
 Test:
(%i20) [sr1^^2-A,sr2^^2-A,sr3^^2-A,sr4^^2-A]$
(%i21) ratsimp(%);
(%o21)
[matrix([0,0],[0,0]),matrix([0,0],[0,0]),matrix([0,0],[0,0]),matrix([0,0],[0,0])]

best

Aleksas Domarkas