Wrong Matrix Af

Af=(({{1, 2}, {3, 4}}));MatrixForm[Af]

SingularValues[Af]

Correct Matrix A

A=(({{1., 2}, {3, 4}})); MatrixForm[A]

{h, s, a} = SingularValues[A]

h

s

a

hanger = Transpose[h];  MatrixForm[hanger]

stretcher = DiagonalMatrix[s]; MatrixForm[stretcher]

MatrixRank[A]

aligner = a; MatrixForm[aligner]

Let us repeat the computation:

Clear[t]

af1 = {Cos[t], Sin[t]};  
    af2 = {Cos[t + (π/2)], Sin[t + (π/2)]};

transedentalEquation = (A . af1) . (A . af2) // 
        Expand

Plot[transedentalEquation, {t, -1, 5}];

sol = FindRoot[transedentalEquation == 0, {t, 0.5}]

{af1 = af1 /. sol, af2 = af2 /. sol} // MatrixForm

a

{xStretcher = Norm[A . af1], yStretcher = Norm[A . af2]}

s

{hf1 = (1/xStretcher) A . af1, 
      hf2 = (1/yStretcher) A . af2}

h