2925.sagews

A=Matrix(ZZ,4,4,[32982266684193100, 1368614777139719, 224591013270052693 , 276460184982223238,1368614777139719,   56791380087354, 9319512049770279, 11471848267545007,224591013270052693 , 9319512049770279 ,1529340971891522140, 1882541434053596358,276460184982223238 ,  11471848267545007, 1882541434053596358, 2317313350044091414])

A._pari_()

[32982266684193100, 1368614777139719, 224591013270052693, 276460184982223238; 1368614777139719, 56791380087354, 9319512049770279, 11471848267545007; 224591013270052693, 9319512049770279, 1529340971891522140, 1882541434053596358; 276460184982223238, 11471848267545007, 1882541434053596358, 2317313350044091414]

pari('qfminim(%s,2,0)'%(A._pari_()))

[0, 0, [;]]

pari(A).qfminim(2,0)

[0, 0, [;]]

B = Matrix(ZZ,4,4,[32982266684193100, 1368614777139719, 224591013270052693, 276460184982223238,1368614777139719,  56791380087354, 9319512049770279, 11471848267545007,224591013270052693, 9319512049770279,1529340971891522140, 1882541434053596358,276460184982223238, 11471848267545007, 1882541434053596358, 2317313350044091414])

pari(B).qfminim(2,0)

[0, 0, [;]]

t = var('t')
f = function('f')
A = matrix([[0,1],[f(t),0]])
exp(A)
diff(exp(A),t)

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rr} -\frac{{\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} + 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right)}{4 \, \sqrt{f\left(t\right)}} + \frac{e^{\sqrt{f\left(t\right)}} D[0]\left(f\right)\left(t\right)}{2 \, \sqrt{f\left(t\right)}} & -\frac{{\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} - 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right)}{4 \, f\left(t\right)} - \frac{{\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} - 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right)}{4 \, f\left(t\right)^{\left(\frac{3}{2}\right)}} + \frac{e^{\sqrt{f\left(t\right)}} D[0]\left(f\right)\left(t\right)}{2 \, f\left(t\right)} \\ -\frac{1}{4} \, {\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} - 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right) + \frac{{\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} - 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right)}{4 \, \sqrt{f\left(t\right)}} + \frac{1}{2} \, e^{\sqrt{f\left(t\right)}} D[0]\left(f\right)\left(t\right) & -\frac{{\left(e^{\left(2 \, \sqrt{f\left(t\right)}\right)} + 1\right)} e^{\left(-\sqrt{f\left(t\right)}\right)} D[0]\left(f\right)\left(t\right)}{4 \, \sqrt{f\left(t\right)}} + \frac{e^{\sqrt{f\left(t\right)}} D[0]\left(f\right)\left(t\right)}{2 \, \sqrt{f\left(t\right)}} \end{array}$$\right)