2×2 のヒルベルト行列の逆行列
$$
\begin{pmatrix}
1 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{3}
\end{pmatrix}
^{-1}\,=\,
\begin{pmatrix}
4 & -6 \\
-6 & 12
\end{pmatrix}
$$
[ [4, -6], [-6, 12] ]
3×3 のヒルベルト行列の逆行列
$$
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5}
\end{pmatrix}
^{-1}\,=\,
\begin{pmatrix}
9 & -36 & 30 \\
-36 & 192 & -180 \\
30 & -180 & 180
\end{pmatrix}
$$
[ [9, -36, 30], [-36, 192, -180], [30, -180, 180] ]
4×4 のヒルベルト行列の逆行列
$$
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}
\end{pmatrix}
^{-1}\,=\,
\begin{pmatrix}
16 & -120 & 240 & -140 \\
-120 & 1200 & -2700 & 1680 \\
240 & -2700 & 6480 & -4200 \\
-140 & 1680 & -4200 & 2800
\end{pmatrix}
$$
[ [16, -120, 240, -140], [-120, 1200, -2700, 1680], [240, -2700, 6480, -4200], [-140, 1680, -4200, 2800] ]
5×5 のヒルベルト行列の逆行列
$$
H_{5} \,=\,
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\
\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9}
\end{pmatrix}
$$
$$
H_{5}^{-1} \,=\,
\begin{pmatrix}
25 & -300 & 1050 & -1400 & 630 \\
-300 & 4800 & -18900 & 26880 & -12600 \\
1050 & -18900 & 79380 & -117600 & 56700 \\
-1400 & 26880 & -117600 & 179200 & -88200 \\
630 & -12600 & 56700 & -88200 & 44100
\end{pmatrix}
$$
[ [25, -300, 1050, -1400, 630], [-300, 4800, -18900, 26880, -12600], [1050, -18900, 79380, -117600, 56700], [-1400, 26880, -117600, 179200, -88200], [630, -12600, 56700, -88200, 44100] ]
6×6 のヒルベルト行列の逆行列
$$
H_{6} \,=\,
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \\
\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} \\
\frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11}
\end{pmatrix}
$$
$$
H_{6}^{-1} \,=\,
\begin{pmatrix}
36 & -630 & 3360 & -7560 & 7560 & -2772 \\
-630 & 14700 & -88200 & 211680 & -220500 & 83160 \\
3360 & -88200 & 564480 & -1411200 & 1512000 & -582120 \\
-7560 & 211680 & -1411200 & 3628800 & -3969000 & 1552320 \\
7560 & -220500 & 1512000 & -3969000 & 4410000 & -1746360 \\
-2772 & 83160 & -582120 & 1552320 & -1746360 & 698544
\end{pmatrix}
$$
[ [36, -630, 3360, -7560, 7560, -2772], [-630, 14700, -88200, 211680, -220500, 83160], [3360, -88200, 564480, -1411200, 1512000, -582120], [-7560, 211680, -1411200, 3628800, -3969000, 1552320], [7560, -220500, 1512000, -3969000, 4410000, -1746360], [-2772, 83160, -582120, 1552320, -1746360, 698544] ]
7×7 のヒルベルト行列の逆行列
$$
H_{7} \,=\,
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} \\
\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} \\
\frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} \\
\frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} & \frac{1}{13}
\end{pmatrix}
$$
$$
H_{7}^{-1} \,=\,
\begin{pmatrix}
49 & -1176 & 8820 & -29400 & 48510 & -38808 & 12012 \\
-1176 & 37632 & -317520 & 1128960 & -1940400 & 1596672 & -504504 \\
8820 & -317520 & 2857680 & -10584000 & 18711000 & -15717240 & 5045040 \\
-29400 & 1128960 & -10584000 & 40320000 & -72765000 & 62092800 & -20180160 \\
48510 & -1940400 & 18711000 & -72765000 & 133402500 & -115259760 & 37837800 \\
-38808 & 1596672 & -15717240 & 62092800 & -115259760 & 100590336 & -33297264 \\
12012 & -504504 & 5045040 & -20180160 & 37837800 & -33297264 & 11099088
\end{pmatrix}
$$
[ [49, -1176, 8820, -29400, 48510, -38808, 12012], [-1176, 37632, -317520, 1128960, -1940400, 1596672, -504504], [8820, -317520, 2857680, -10584000, 18711000, -15717240, 5045040], [-29400, 1128960, -10584000, 40320000, -72765000, 62092800, -20180160], [48510, -1940400, 18711000, -72765000, 133402500, -115259760, 37837800], [-38808, 1596672, -15717240, 62092800, -115259760, 100590336, -33297264], [12012, -504504, 5045040, -20180160, 37837800, -33297264, 11099088] ]
8×8 のヒルベルト行列の逆行列
$$
H_{8} \,=\,
\begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} \\
\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} \\
\frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} & \frac{1}{13} \\
\frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} & \frac{1}{13} & \frac{1}{14} \\
\frac{1}{8} & \frac{1}{9} & \frac{1}{10} & \frac{1}{11} & \frac{1}{12} & \frac{1}{13} & \frac{1}{14} & \frac{1}{15}
\end{pmatrix}
$$
$$
H_{8}^{-1} \,=\,
\begin{pmatrix}
64 & -2016 & 20160 & -92400 & 221760 & -288288 & 192192 & -51480 \\
-2016 & 84672 & -952560 & 4656960 & -11642400 & 15567552 & -10594584 & 2882880 \\
20160 & -952560 & 11430720 & -58212000 & 149688000 & -204324120 & 141261120 & -38918880 \\
-92400 & 4656960 & -58212000 & 304920000 & -800415000 & 1109908800 & -776936160 & 216216000 \\
221760 & -11642400 & 149688000 & -800415000 & 2134440000 & -2996753760 & 2118916800 & -594594000 \\
-288288 & 15567552 & -204324120 & 1109908800 & -2996753760 & 4249941696 & -3030051024 & 856215360 \\
192192 & -10594584 & 141261120 & -776936160 & 2118916800 & -3030051024 & 2175421248 & -618377760 \\
-51480 & 2882880 & -38918880 & 216216000 & -594594000 & 856215360 & -618377760 & 176679360
\end{pmatrix}
$$
[ [64, -2016, 20160, -92400, 221760, -288288, 192192, -51480], [-2016, 84672, -952560, 4656960, -11642400, 15567552, -10594584, 2882880], [20160, -952560, 11430720, -58212000, 149688000, -204324120, 141261120, -38918880], [-92400, 4656960, -58212000, 304920000, -800415000, 1109908800, -776936160, 216216000], [221760, -11642400, 149688000, -800415000, 2134440000, -2996753760, 2118916800, -594594000], [-288288, 15567552, -204324120, 1109908800, -2996753760, 4249941696, -3030051024, 856215360], [192192, -10594584, 141261120, -776936160, 2118916800, -3030051024, 2175421248, -618377760], [-51480, 2882880, -38918880, 216216000, -594594000, 856215360, -618377760, 176679360] ]
9×9 のヒルベルト行列の逆行列
[ [81, -3240, 41580, -249480, 810810, -1513512, 1621620, -926640, 218790], [-3240, 172800, -2494800, 15966720, -54054000, 103783680, -113513400, 65894400, -15752880], [41580, -2494800, 38419920, -256132800, 891891000, -1748106360, 1942340400, -1141620480, 275675400], [-249480, 15966720, -256132800, 1756339200, -6243237000, 12430978560, -13984850880, 8302694400, -2021619600], [810810, -54054000, 891891000, -6243237000, 22545022500, -45450765360, 51648597000, -30918888000, 7581073500], [-1513512, 103783680, -1748106360, 12430978560, -45450765360, 92554285824, -106051785840, 63930746880, -15768632880], [1621620, -113513400, 1942340400, -13984850880, 51648597000, -106051785840, 122367445200, -74205331200, 18396738360], [-926640, 65894400, -1141620480, 8302694400, -30918888000, 63930746880, -74205331200, 45229916160, -11263309200], [218790, -15752880, 275675400, -2021619600, 7581073500, -15768632880, 18396738360, -11263309200, 2815827300] ]
10×10 のヒルベルト行列の逆行列
[ [100, -4950, 79200, -600600, 2522520, -6306300, 9609600, -8751600, 4375800, -923780], [-4950, 326700, -5880600, 47567520, -208107900, 535134600, -832431600, 770140800, -389883780, 83140200], [79200, -5880600, 112907520, -951350400, 4281076800, -11237826600, 17758540800, -16635041280, 8506555200, -1829084400], [-600600, 47567520, -951350400, 8245036800, -37875637800, 101001700800, -161602721280, 152907955200, -78843164400, 17071454400], [2522520, -208107900, 4281076800, -37875637800, 176752976400, -477233036280, 771285715200, -735869534400, 382086104400, -83223340200], [-6306300, 535134600, -11237826600, 101001700800, -477233036280, 1301544644400, -2121035716800, 2037792556800, -1064382719400, 233025352560], [9609600, -832431600, 17758540800, -161602721280, 771285715200, -2121035716800, 3480673996800, -3363975014400, 1766086882560, -388375587600], [-8751600, 770140800, -16635041280, 152907955200, -735869534400, 2037792556800, -3363975014400, 3267861442560, -1723286307600, 380449555200], [4375800, -389883780, 8506555200, -78843164400, 382086104400, -1064382719400, 1766086882560, -1723286307600, 912328045200, -202113826200], [-923780, 83140200, -1829084400, 17071454400, -83223340200, 233025352560, -388375587600, 380449555200, -202113826200, 44914183600] ]
11×11 のヒルベルト行列の逆行列
[ [121, -7260, 141570, -1321320, 6936930, -22198176, 44924880, -57760560, 45727110, -20323160, 3879876], [-7260, 580800, -12741300, 126846720, -693693000, 2283240960, -4717112400, 6161126400, -4938527880, 2217072000, -426786360], [141570, -12741300, 298146420, -3091888800, 17391874500, -58436698320, 122644922400, -162191652480, 131319945900, -59445243000, 11523231720], [-1321320, 126846720, -3091888800, 32980147200, -189378189000, 646410885120, -1373623130880, 1834895462400, -1498020123600, 682858176000, -133157344320], [6936930, -693693000, 17391874500, -189378189000, 1104706102500, -3817864290240, 8194910724000, -11038043016000, 9074544979500, -4161167010000, 815588733960], [-22198176, 2283240960, -58436698320, 646410885120, -3817864290240, 13327817158656, -28846085748480, 39125617090560, -32357234669760, 14913622563840, -2936119442256], [44924880, -4717112400, 122644922400, -1373623130880, 8194910724000, -28846085748480, 62869674067200, -85781362867200, 71305757883360, -33011924946000, 6524709871680], [-57760560, 6161126400, -162191652480, 1834895462400, -11038043016000, 39125617090560, -85781362867200, 117643011932160, -98227319533200, 45653946624000, -9054699413760], [45727110, -4938527880, 131319945900, -1498020123600, 9074544979500, -32357234669760, 71305757883360, -98227319533200, 82337606079300, -38401626978000, 7639902630360], [-20323160, 2217072000, -59445243000, 682858176000, -4161167010000, 14913622563840, -33011924946000, 45653946624000, -38401626978000, 17965673440000, -3584151851280], [3879876, -426786360, 11523231720, -133157344320, 815588733960, -2936119442256, 6524709871680, -9054699413760, 7639902630360, -3584151851280, 716830370256] ]
12×12 のヒルベルト行列の逆行列
[ [144, -10296, 240240, -2702700, 17297280, -68612544, 176432256, -299304720, 332560800, -232792560, 93117024, -16224936], [-10296, 981552, -25765740, 309188880, -2061259200, 8409937536, -22076086032, 38044955520, -42800574960, 30263032800, -12206089896, 2141691552], [240240, -25765740, 721440720, -9018009000, 61837776000, -257554337040, 686811565440, -1198416098880, 1361836476000, -970938969000, 394350596640, -69604975440], [-2702700, 309188880, -9018009000, 115945830000, -811620810000, 3434057827200, -9271956133440, 16342037712000, -18725251545000, 13443770340000, -5492740453200, 974469656160], [17297280, -2061259200, 61837776000, -811620810000, 5771525760000, -24725216355840, 67432408243200, -119841609888000, 138278780640000, -99868008240000, 41012462050560, -7308522421200], [-68612544, 8409937536, -257554337040, 3434057827200, -24725216355840, 106992754412544, -294230074634496, 526565596677120, -611192210428800, 443680271274240, -183018111900624, 32742180446976], [176432256, -22076086032, 686811565440, -9271956133440, 67432408243200, -294230074634496, 814790975910912, -1466861305029120, 1711338189200640, -1247850762958800, 516757021837056, -92769511266432], [-299304720, 38044955520, -1198416098880, 16342037712000, -119841609888000, 526565596677120, -1466861305029120, 2654320456719360, -3110531785218000, 2276990587872000, -946216088737920, 170392979877120], [332560800, -42800574960, 1361836476000, -18725251545000, 138278780640000, -611192210428800, 1711338189200640, -3110531785218000, 3659449159080000, -2688113888460000, 1120519052452800, -202341663604080], [-232792560, 30263032800, -970938969000, 13443770340000, -99868008240000, 443680271274240, -1247850762958800, 2276990587872000, -2688113888460000, 1980715496760000, -827939077645680, 149882713780800], [93117024, -12206089896, 394350596640, -5492740453200, 41012462050560, -183018111900624, 516757021837056, -946216088737920, 1120519052452800, -827939077645680, 346945899203904, -62950739787936], [-16224936, 2141691552, -69604975440, 974469656160, -7308522421200, 32742180446976, -92769511266432, 170392979877120, -202341663604080, 149882713780800, -62950739787936, 11445589052352] ]
