mat0[u_,v_]:=Module[{eve,mm},
eve=Norm[#]&/@{u,v};
mm=MapThread[Divide,{{u,v},eve}];
mm={mm[[1]],{{1,-1},{1,1}}.mm[[1]]};
Inverse[mm].DiagonalMatrix[eve].mm
];
mat1[u_,v_]:=Module[{eve,mm},
eve=Norm[#]&/@{u,v};
mm=MapThread[Divide,{{u,v},eve}];
Inverse[mm].DiagonalMatrix[eve].mm
];
mat2[u_,v_]:=Module[{eve,mm},
eve=Norm[#]&/@{u,v};
eve[[1]]=-eve[[1]];
mm=MapThread[Divide,{{u,v},eve}];
Inverse[mm].DiagonalMatrix[eve].mm
];
mat3[u_,v_]:=Module[{eve,mm},
eve=Norm[#]&/@{u,v};
eve=-eve;
mm=MapThread[Divide,{{u,v},eve}];
Inverse[mm].DiagonalMatrix[eve].mm
];
man[mat_]:=Module[{Id,T,smat,drawmat,def,lines},
lines[u_,v_]:=Graphics[{Line[{{0,0},u}],Line[{{0,0},v}]},PlotRange->{{-1,1},{-1,1}}];
Id=IdentityMatrix[2];
T=Transpose;Id={{1,0},{0,1}};lines[u_,v_]:=Graphics[{Line[{{0,0},u}],Line[{{0,0},v}]},PlotRange->{{-1,1},{-1,1}}];
smat[u_,v_]:=Inverse[2Id+mat[u,v]+T[mat[u,v]]] 2Id;
drawmat[A_]:=Module[{f,points,targets,arrows,plot,nn},
nn=13;
f[x_]={Cos[x],Sin[x]};
points=f[#]&/@Table[x,{x,0,2 Pi,2 Pi/nn}];
targets=#.A&/@points;
arrows=MapThread[Arrow[{#1,#2}]&,{points,targets}];
plot=ParametricPlot[{f[x],f[x].A},{x,0,2Pi}];
Show[Graphics[arrows],plot]
];
def[u_,v_]:=And@@(Thread[Eigenvalues[mat[u,v]]>{0,0}]);
Manipulate[Show[
lines[u,v],
drawmat[smat[u,v]],
Graphics[Text["Condition Number   "<>(ToString[LinearAlgebra`MatrixConditionNumber[mat[u,v]]]),{-.5,-1.2}]],
PlotRange->{{-2,2},{-2,2}}],
{{u,{1,1}},Locator},{{v,{-1.,1}},Locator}]
]

Positive Definite symmetric

man[mat0]

Positive Definite arbitary

man[mat1]

DynamicModule[{u={1,1},v={-1.`,1}},Show[lines$68755[u,v],drawmat$68755[smat$68755[u,v]],Graphics[Text["Condition Number   "<>ToString[LinearAlgebra`MatrixConditionNumber[mat1[u,v]]],{-0.5`,-1.2`}]],PlotRange->{{-2,2},{-2,2}}]]

Indefinite Regularization matrices

man[mat2]

DynamicModule[{u={1,1},v={-1.`,1}},Show[lines$72168[u,v],drawmat$72168[smat$72168[u,v]],Graphics[Text["Condition Number   "<>ToString[LinearAlgebra`MatrixConditionNumber[mat2[u,v]]],{-0.5`,-1.2`}]],PlotRange->{{-2,2},{-2,2}}]]

Negative Definite regularization matrices

man[mat3]

DynamicModule[{u={1,1},v={-1.`,1}},Show[lines$72111[u,v],drawmat$72111[smat$72111[u,v]],Graphics[Text["Condition Number   "<>ToString[LinearAlgebra`MatrixConditionNumber[mat3[u,v]]],{-0.5`,-1.2`}]],PlotRange->{{-2,2},{-2,2}}]]