mapBall[A_] := Module[{},
Print["A = ", MatrixForm[A]];
Print["Determinant of A = ", Det[A]];
{evals, evecs} = Eigensystem@N[A];
Print["Eigenvalues of A are ", evals];
If[Abs@Im[evals[[1]] ] > 10^-5, realEvecs = False;
Print["Eigenvalues have magnitude ", Abs[evals[[1]] ] ],realEvecs = True;
{evec1, evec2} = evecs
];
ATA = Transpose[A].A;
Print["(A^T)A = ", MatrixForm[ATA]];
{{s1squared, s2squared}, {v1, v2}} = Eigensystem[ATA//N];
{s1, s2} = Sqrt[{s1squared,s2squared}];
Print["Singular values of A are ", {s1, s2}];
dt = Pi/24;
unitCircle = Line[Table[{Cos[t], Sin[t]}, {t, 0, 2Pi, dt}] ];
pieSlices = Table[{Hue[t/(2Pi)],Polygon[{{Cos[t], Sin[t]}, {Cos[t+dt], Sin[t+dt]}, {0,0}}]}, {t, 0, 2Pi-dt, dt}];
pieSlicesA = Table[{Hue[t/(2Pi)],Polygon[{A.{Cos[t], Sin[t]}, A.{Cos[t+dt], Sin[t+dt]}, {0,0}}]}, {t, 0, 2Pi-dt, dt}];
axesN = {Black,Thickness[.01],Line[{{0,0}, v1}], Line[{{0,0}, v2}]};
axesAN = {Black,Thickness[0.01],Line[{{0,0}, A.v1}], Line[{{0,0}, A.v2}]};
If[realEvecs,
evecLines = {White,Thickness[.008],Line[{evec1, -evec1}], Line[{evec2, -evec2}]},
(*else*) evecLines = {}];
p1 = Graphics[{pieSlices, evecLines, axesN}];
p2 = Graphics[{pieSlicesA, unitCircle, evecLines,axesAN}];
GraphicsRow[{p1, p2}]
]

mapBall[{{1,1}, {0,1}}]