[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/17110#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/17110","headline":"Hilbert -Matrix -Wikipedia","name":"Hilbert -Matrix -Wikipedia","description":"before-content-x4 \u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9 \u6ce8\u6587 n \u2265 \u521d\u3081 {displaystyle ngeq 1} after-content-x4 \u6b21\u306e\u6b63\u65b9\u5f62\u3001\u5bfe\u79f0\u7684\u3067\u6b63\u306e\u660e\u78ba\u306a\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u3059\u3002 Hn= (11213\u22ef1n121314\u22ef1n+1131415\u22ef1n+2\u22ee\u22ee\u22ee\u22f1\u22ee1n1n+11n+2\u22ef12n\u22121){displaystyle h_ {n} = {begin {pmatrix} 1\uff06{frac {1}","datePublished":"2020-02-17","dateModified":"2020-02-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/17110","wordCount":4619,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4 \u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9 \u6ce8\u6587 n \u2265 \u521d\u3081 {displaystyle ngeq 1}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u6b21\u306e\u6b63\u65b9\u5f62\u3001\u5bfe\u79f0\u7684\u3067\u6b63\u306e\u660e\u78ba\u306a\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3067\u3059\u3002 Hn= (11213\u22ef1n121314\u22ef1n+1131415\u22ef1n+2\u22ee\u22ee\u22ee\u22f1\u22ee1n1n+11n+2\u22ef12n\u22121){displaystyle h_ {n} = {begin {pmatrix} 1\uff06{frac {1} {2}}\uff06{frac {1} {3}}\uff06{frac {1} {n}}} \\ {frac {1} {1} {} {} {{1} {1} {1} {1} {1} } {4}}\uff06cdots\uff06{frac {1} {n+1}} \\ {frac {1} {3}}\uff06{frac {1} {4}}\uff06{frac {1} {5}}\uff06cdots\uff06{frac {1} {n+2} \\ vd} \\ vd} \\ vd} \\ vdots \\ {frac {1} {n}}\uff06{frac {1} {n+1}}\uff06{frac {1} {n+2}}\uff06{frac {1} {2n-1}} end {pmatrix}}}}}}}}} \u3001 \u3057\u305f\u304c\u3063\u3066\u3001\u500b\u3005\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u306f\u901a\u904e\u3057\u3066\u3044\u307e\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4h ij= 1i+j\u22121{displaystyle h_ {ij} = {frac {1} {i+j-1}}}}}} \u4e0e\u3048\u3089\u308c\u305f\u3002\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u306f\u3001\u7a4d\u5206\u3092\u4f34\u3046\u6b74\u53f2\u7684\u30a2\u30af\u30bb\u30b9\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002 h ij= \u222b 01\u30d0\u30c4 i+j\u22122d \u30d0\u30c4 {displaystyle h_ {ij} = int _ {0}^{1} x^{i+j-2}\u3001dx} \u3002 \u3053\u308c\u306f\u30011894\u5e74\u306bLegendre Polynomas\u306e\u7406\u8ad6\u306b\u95a2\u9023\u3057\u3066\u3001\u30c9\u30a4\u30c4\u306e\u6570\u5b66\u8005David Hilbert\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3057\u305f\u3002\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306f\u6b63\u3067\u3042\u308b\u305f\u3081\u3001\u305d\u306e\u9006\u304c\u5b58\u5728\u3057\u307e\u3059\u3002 H.\u3053\u308c\u3089\u306e\u4fc2\u6570\u3092\u4f7f\u7528\u3057\u305f\u65b9\u7a0b\u5f0f\u306e\u7dda\u5f62\u30b7\u30b9\u30c6\u30e0\u306f\u3001\u7c21\u5358\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u307e\u305f\u306f\u95a2\u9023\u3059\u308b\u65b9\u7a0b\u5f0f\u306e\u30b7\u30b9\u30c6\u30e0\u306f\u6bd4\u8f03\u7684\u6761\u4ef6\u304c\u52a3\u3063\u3066\u3044\u307e\u3059\u3002 n {displaystyle n} \u306f\u3002\u72b6\u614b\u306e\u72b6\u614b\u306f\u6307\u6570\u95a2\u6570\u7684\u306b\u5897\u52a0\u3057\u307e\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4n {displaystyle n} ;\u306e\u6761\u4ef6\u756a\u53f7 h 3{displaystyle h_ {3}} 526.16\uff08Frobenius Standard\uff09\u3001\u305d\u308c\u3067\u3059 h 4{displaystyle h_ {4}} 15,613.8\u3002\u3053\u308c\u306f\u3001\u9006\u6570\uff08\u65b9\u7a0b\u5f0f\u30b7\u30b9\u30c6\u30e0\u306e\u6eb6\u89e3\uff09\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306b\u5927\u304d\u306a\u6570\u304c\u767a\u751f\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002 n {displaystyle n} \u306f\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30d2\u30eb\u30d9\u30eb\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306f\u53e4\u5178\u7684\u306a\u3082\u306e\u3067\u3059 \u30c6\u30b9\u30c8\u30b1\u30fc\u30b9 \u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u53cd\u8ee2\u307e\u305f\u306f\u7dda\u5f62\u65b9\u7a0b\u5f0f\u30b7\u30b9\u30c6\u30e0\u306e\u89e3\u50cf\u5ea6\u306e\u305f\u3081\u306e\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u5834\u5408\u3001\u4f8b\u3048\u3070B. Gauss\u30e1\u30bd\u30c3\u30c9\u3001LR\u5206\u89e3\u3001\u8edf\u9aa8\u5206\u89e3\u306a\u3069\u3002\u9006\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u3059\u3079\u3066\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u306f\u3001\u4ea4\u4e92\u306e\u5146\u5019\u3092\u6301\u3064\u6574\u6570\u3067\u3059\u3002 \u30d2\u30eb\u30d9\u30eb\u30c8\u884c\u5217\u306e\u9006\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u306f\u3001\u9589\u3058\u305f\u5f0f\u3067\u76f4\u63a5\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002 \uff08 Hn\u22121)i,j =  (\u22121)i+j(i+j\u22121) (n+i\u22121)!(n+j\u22121)!((i\u22121)!(j\u22121)!)2(n\u2212i)!(n\u2212j)!{displaystyle\uff08h_ {n}^{-1}\uff09_ {i\u3001j} = {frac {\uff08-1\uff09^{i+j}} {\uff08i+j-1\uff09}}}} {\uff08n+i-1\uff09 \u3001 \u4e8c\u9805\u4fc2\u6570\u3092\u901a\u3058\u3066\u8868\u73fe\u3067\u304d\u308b\u3082\u306e\uff1a \uff08 Hn\u22121)i,j =  \uff08  –  \u521d\u3081 )i+j\uff08 \u79c1 + j  –  \u521d\u3081 \uff09\uff09 (n+i\u22121n\u2212j)(n+j\u22121n\u2212i)(i+j\u22122i\u22121)2{displaystyle\uff08h_ {n}^{ –  1}\uff09_ {i\u3001j} =\uff08-1\uff09^{i+j}\uff08i+j-1\uff09{n+i-1 choose n-j} {n-i} {i+j-2 choice i-1}^{2}} \u3002 \u7279\u5225\u306a\u5834\u5408 \u79c1 = j = \u521d\u3081 {displaystyle i = j = 1} \u3053\u308c\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \uff08 Hn\u22121)1,1 =  n2{displaystyle\uff08h_ {n}^{ –  1}\uff09_ {1.1} = n^{2}}}} \u3002 \u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u9006\u3092\u6b63\u78ba\u306b\u8a08\u7b97\u3067\u304d\u308b\u3053\u3068\u306f\u7279\u306b\u4fbf\u5229\u3067\u3059\u3002 B.\u30c6\u30b9\u30c8\u3067\u306f\u3001LR\u307e\u305f\u306f\u80c6\u56a2\u5206\u89e3\u3092\u5099\u3048\u305f\u30d2\u30eb\u30d9\u30eb\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u6570\u5024\u53cd\u8ee2\u306e\u7d50\u679c\u3002\u3053\u308c\u306f\u3001\u4e38\u3081\u30a8\u30e9\u30fc\u306b\u3088\u3063\u3066\u81ea\u7136\u306b\u640d\u306a\u308f\u308c\u307e\u3059\u3002 \u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u9006\u306e\u6c7a\u5b9a\u8981\u56e0\u306f\u3001\u6b21\u306e\u5f0f\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u6b63\u78ba\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002  Hn\u22121= \u220fk=1n\u22121\uff08 2 k + \u521d\u3081 \uff09\uff09 (2kk)2{displaystyle det h_ {n}^{ –  1} = prod _ {k = 1}^{n-1}\uff082k+1\uff09{2k choose k}^{2}} \u30d2\u30eb\u30d9\u30eb\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u6c7a\u5b9a\u8981\u56e0\u3068\u3057\u3066\u3001\u9006\u306e\u7d50\u679c\u306e\u76f8\u4e92\u5024\u306f  h n= \uff08  h n\u22121\uff09\uff09 \u22121{displaystyle it h_ {n} =\uff08it h_ {n}^{ –  1}\uff09^{-1}} \u3002\u9006\u306e\u6c7a\u5b9a\u8981\u56e0 \u521d\u3081 \u2264 n \u2264 5 {displaystyle 1leq nleq 5} 1\u300112\u30012160\u30016048000\u3001266716800000\u3067\u3059\uff08\u7d50\u679c A005249 OEIS\u3067\uff09\u3002 \u4e0a\u8a18\u306e\u5f0f\u306f\u3001\u5834\u5408\u306b\uff08\u6b63\u78ba\u306a\uff09\u9006\u306b\u306a\u308a\u307e\u3059 n = 2 \u3001 3 \u3001 4 \u3001 5 {displaystyle n = 2,3,4,5} \uff1a H2\u22121 =  (4\u22126\u2212612){displaystyle h_ {2}^{ –  1} = {begin {pmatrix} 4\uff06-6 \\ -6\uff0612end {pmatrix}}}} \u3001 H3\u22121 =  (9\u22123630\u221236192\u221218030\u2212180180){displaystyle h_ {3}^{ –  1} = {begin {pmatrix} 9\uff06-36\uff0630 \\ -36\uff06192\uff06-180 \\ 30\uff06-180\uff06180end {pmatrix}}}} \u3001 H4\u22121 =  (16\u2212120240\u2212140\u22121201200\u221227001680240\u221227006480\u22124200\u22121401680\u221242002800){displaystyle h_ {4}^{-1} = {begin {pmatrix} 16\uff06-120\uff06240\uff06-140 \\ -120\uff06-2700\uff061680 \\ 240\uff06-2700\uff066480\uff06-4200 \\ -140\uff061680\uff06-4200\uff062800END}} { \u3001 H5\u22121 =  (25\u22123001050\u22121400630\u22123004800\u22121890026880\u2212126001050\u22121890079380\u221211760056700\u2212140026880\u2212117600179200\u221288200630\u22121260056700\u22128820044100){displaystyle h_ {5}^{-1} = {begin {pmatrix} 25\uff06-300\uff061050\uff06-1400\uff06630 \\ -300\uff064800\uff06-18900\uff0626880\uff06-12600 \\ 1050\uff06-18900\uff0679380\uff06-117600\uff0656700\uff06-1400\uff06-1400 \uff06-88200 \\ 630\uff06-12600\uff0656700\uff06-88200\uff0644100END {pmatrix}}} \u3002 Hilbert\uff08\u305d\u3057\u3066\u3082\u3061\u308d\u3093\u4ed6\u306e\u3059\u3079\u3066\u306e\uff09\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3092\u5b9f\u9a13\u3059\u308b\u305f\u3081\u306b\u306f\u3001Matlab\u3001Maple\u3001Gnu Octave\u3001Mathematica\u306a\u3069\u306e\u6700\u65b0\u306e\u6570\u5b66\u30bd\u30d5\u30c8\u30a6\u30a7\u30a2\u30d1\u30c3\u30b1\u30fc\u30b8\u304c\u5f79\u7acb\u3061\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001Mathematica\u3067\u306f\u3001\u6700\u5f8c\u306e\u9006\u306f\u6b21\u306e\u30b3\u30de\u30f3\u30c9\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002 \u9006 n = 5 {displaystyle n = 5} \u8a08\u7b97\uff1a  [1]\uff1a= Inverse [hilbertmatrix [5]] \/\/ TraditionalForm  \u30d2\u30eb\u30d0\u30fc\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u306e\u9069\u6027\u304c\u60aa\u3044\u3053\u3068\u306f\u3001\u5b9f\u8cea\u7684\u306b\u30e9\u30a4\u30f3\uff08\u304a\u3088\u3073\u305d\u306e\u7d50\u679c\u3082\u5217\uff09\u30d9\u30af\u30c8\u30eb\u3092\u610f\u5473\u3057\u307e\u3059 \u901f\u3044 \u7dda\u5f62\u4f9d\u5b58\u3067\u3059\u3002\u5e7e\u4f55\u5b66\u7684\u306b\u3001u\u3002\u30e9\u30a4\u30f3\u30d9\u30af\u30c8\u30eb\u9593\u306e\u89d2\u5ea6\u306f\u975e\u5e38\u306b\u5c0f\u3055\u304f\u3001\u6700\u5f8c\u306e\u30e9\u30a4\u30f3\u30d9\u30af\u30c8\u30eb\u9593\u306e\u6700\u5c0f\u306e\u3082\u306e\u3067\u3042\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u3002 z\u3002 B.\u6700\u5f8c\u306e\u7dda\u3068\u6700\u5f8c\u304b\u30892\u756a\u76ee\u306e\u30e9\u30a4\u30f3\u30d9\u30af\u30c8\u30eb\u306e\u9593\u306e\u89d2\u5ea6 h 4{displaystyle h_ {4}} 3\u00b0\u3088\u308a\u5c0f\u3055\u3044\uff08\u5f13\u306e\u4e2d\uff1a\u3088\u308a\u5c0f\u3055\u3044 \u03c060 {displaystyle {frac {pi} {60}}} \uff09\u3002\u5927\u304d\u306a\u3082\u306e\u3067 n {displaystyle n} \u89d2\u5ea6\u306f\u3055\u3089\u306b\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u306e\u6700\u521d\u306e\u7dda\u30d9\u30af\u30c8\u30eb\u9593\u306e\u89d2\u5ea6 h 3{displaystyle h_ {3}} \u307e\u305f\u3001\u4ed6\u306e2\u3064\u306e\u30e9\u30a4\u30f3\u30d9\u30af\u30c8\u30eb\u306b\u3088\u3063\u3066\u30af\u30e9\u30f3\u30d7\u3055\u308c\u308b\u30ec\u30d9\u30eb\u306f1.3\u00b0\u3088\u308a\u308f\u305a\u304b\u306b\u5c0f\u3055\u304f\u3001\u4ed6\u306e2\u3064\u306e\u30e9\u30a4\u30f3\u30d9\u30af\u30c8\u30eb\u306e\u5bfe\u5fdc\u3059\u308b\u89d2\u5ea6\u306f\u3055\u3089\u306b\u5c0f\u3055\u304f\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u306e\u89d2\u5ea6\u306f\u3001\u3088\u308a\u5927\u304d\u306a\u89d2\u5ea6\u3082\u3042\u308a\u307e\u3059 n {displaystyle n} \u3055\u3089\u306b\u5c0f\u3055\u304f\u3002       (adsbygoogle = window.adsbygoogle || 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