1059:
458:
1054:{\displaystyle {\begin{aligned}R^{2}&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+s^{2}&AB+BD+2sB\\CA+DC+2sC&CB+D^{2}+2sD+s^{2}\end{pmatrix}}\\&={\frac {1}{t^{2}}}{\begin{pmatrix}A^{2}+BC+2sA+AD-BC&AB+BD+2sB\\AC+CD+2sC&BC+D^{2}+2sD+AD-BC\end{pmatrix}}\\&={\frac {1}{A+D+2s}}{\begin{pmatrix}A(A+D+2s)&B(A+D+2s)\\C(A+D+2s)&D(A+D+2s)\end{pmatrix}}=M.\end{aligned}}}
2056:
1576:
1386:
447:
1226:
1945:
1936:
2224:
1470:
320:
463:
1266:
1840:
1758:
175:
333:
1137:
2128:
1700:
2297:
2152:
2411:
260:
1886:
1783:
118:
2051:{\displaystyle {\frac {1}{t}}{\begin{pmatrix}s&r\\r&-s\end{pmatrix}}{\text{ and }}{\begin{pmatrix}\pm 1&0\\0&\pm 1\end{pmatrix}},}
1571:{\displaystyle R={\begin{pmatrix}0&0\\c&0\end{pmatrix}}\quad {\text{and}}\quad R={\begin{pmatrix}0&b\\0&0\end{pmatrix}},}
1705:
1381:{\displaystyle R=\pm {\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}=\pm {\frac {1}{t}}\left(M+sI\right),}
442:{\displaystyle R={\frac {1}{t}}{\begin{pmatrix}A+s&B\\C&D+s\end{pmatrix}}={\frac {1}{t}}\left(M+sI\right).}
2336:
1860:. This, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both
2415:
1130:
is nonzero, the general formula above will give only two distinct solutions, corresponding to the two signs of
104:
51:
206:
2077:
1221:{\displaystyle R=\pm {\frac {1}{t}}{\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\pm {\frac {1}{t}}M,}
2344:
2301:
24:
115:
The following is a general formula that applies to almost any 2 × 2 matrix. Let the given matrix be
2386:
Mitchell, Douglas W. (November 2003), "87.57 Using
Pythagorean triples to generate square roots of
2434:
2361:
2318:
1664:
2276:
2457:
1610:
2424:
2353:
2310:
1939:
1622:
2389:
2373:
1621:, then if it is not the identity matrix, its determinant is zero, and its trace equals its
2369:
1881:
43:
1443:, so that both the trace and the determinant of the matrix are zero. In this case, if
2451:
2357:
2314:
1931:{\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)}
50:
with itself. In general, there can be zero, two, four, or even an infinitude of
2219:{\displaystyle R={\begin{pmatrix}a&0\\{\frac {C}{a+d}}&d\end{pmatrix}}.}
1896:
226:
2429:
1877:
1252:
100:
1071:
is a real matrix; this will be the case, in particular, if the determinant
315:{\displaystyle s=\pm {\sqrt {\delta }},\qquad t=\pm {\sqrt {\tau +2s}}.}
2438:
2365:
2322:
1625:, which (excluding the zero matrix) is 1. Then the above formula has
1835:{\displaystyle R={\begin{pmatrix}a&0\\0&d\end{pmatrix}},}
1753:{\displaystyle \pm {\sqrt {r}}\exp \left({\tfrac {1}{2}}A\right)}
1657:
can be expressed as real multiple of the exponent of some matrix
1102:. Then the formula above will provide four distinct square roots
170:{\displaystyle M={\begin{pmatrix}A&B\\C&D\end{pmatrix}},}
61:
Square roots that are not the all-zeros matrix come in pairs: if
2273:
Levinger, Bernard W. (September 1980), "The square root of a
2244:= 0, and four otherwise. A similar formula can be used when
2167:
2008:
1964:
1798:
1731:
1534:
1485:
1294:
1239:
The formula also gives only two distinct solutions if
1165:
911:
698:
505:
358:
133:
2392:
2279:
2155:
2080:
1948:
1889:
1786:
1708:
1667:
1473:
1269:
1140:
461:
336:
263:
121:
1463:= 0), then the null matrix is also a square root of
2405:
2291:
2218:
2122:
2050:
1930:
1834:
1752:
1694:
1570:
1380:
1220:
1053:
441:
314:
193:may be real or complex numbers. Furthermore, let
169:
107:has precisely one positive-definite square root.
1588:are arbitrary real or complex values. Otherwise
2335:Harkin, Anthony A.; Harkin, Joseph B. (2004),
8:
2228:This formula will provide two solutions if
1263:be zero. In that case, the two roots are
2428:
2397:
2391:
2337:"Geometry of generalized complex numbers"
2278:
2182:
2162:
2154:
2111:
2098:
2085:
2079:
2003:
1998:
1959:
1949:
1947:
1894:
1888:
1793:
1785:
1780:= 0), one can use the simplified formula
1730:
1712:
1707:
1666:
1529:
1517:
1480:
1472:
1343:
1289:
1279:
1268:
1202:
1160:
1150:
1139:
1078:The general case of this formula is when
906:
879:
825:
705:
693:
685:
676:
652:
627:
546:
512:
500:
492:
483:
470:
462:
460:
404:
353:
343:
335:
293:
273:
262:
128:
120:
1760:. In this case the square root is real.
1255:), in which case one of the choices for
58:can be obtained by an explicit formula.
2265:
99:A 2×2 matrix with two distinct nonzero
1419:The formula above fails completely if
1098:is nonzero for each choice of sign of
7:
1702:, then two of its square roots are
1106:, one for each choice of signs for
2123:{\displaystyle r^{2}+s^{2}=t^{2}.}
2074:are any complex numbers such that
14:
2133:Matrix with one off-diagonal zero
1895:
1067:may have complex entries even if
1942:rational square roots given by
1232:is any square root of the trace
2149:are not both zero, one can use
1522:
1516:
283:
54:. In many cases, such a matrix
2358:10.1080/0025570X.2004.11953236
2315:10.1080/0025570X.1980.11976858
1689:
1683:
1027:
1006:
998:
977:
967:
946:
938:
917:
1:
1596:Formulas for special matrices
1118:Special cases of the formula
1427:are both zero; that is, if
18:square root of a 2×2 matrix
2474:
1695:{\displaystyle M=r\exp(A)}
1259:will make the denominator
2430:10.1017/S0025557200173723
2292:{\displaystyle 2\times 2}
1876:Because it has duplicate
1447:is the null matrix (with
103:has four square roots. A
73:is also a square root of
2416:The Mathematical Gazette
1868:are zero, respectively.
105:positive-definite matrix
1641:as two square roots of
1251:(the case of duplicate
1126:is zero, but the trace
2407:
2293:
2220:
2124:
2052:
1932:
1836:
1772:is diagonal (that is,
1754:
1696:
1572:
1408:is any square root of
1392:is the square root of
1382:
1222:
1055:
451:Indeed, the square of
443:
326:≠ 0, a square root of
316:
171:
2408:
2406:{\displaystyle I_{2}}
2294:
2221:
2125:
2053:
1933:
1837:
1755:
1697:
1573:
1383:
1223:
1056:
444:
317:
172:
2390:
2345:Mathematics Magazine
2302:Mathematics Magazine
2277:
2153:
2078:
1946:
1938:has infinitely many
1887:
1784:
1706:
1665:
1592:has no square root.
1471:
1267:
1138:
459:
334:
261:
119:
65:is a square root of
52:square-root matrices
2256:are not both zero.
1467:, as is any matrix
1122:If the determinant
2403:
2289:
2216:
2207:
2120:
2048:
2039:
1992:
1928:
1922:
1921:
1832:
1823:
1750:
1740:
1692:
1649:Exponential matrix
1568:
1559:
1510:
1378:
1331:
1218:
1190:
1051:
1049:
1032:
863:
660:
439:
395:
312:
167:
158:
2198:
2001:
1957:
1739:
1717:
1611:idempotent matrix
1601:Idempotent matrix
1520:
1351:
1287:
1210:
1158:
904:
691:
498:
412:
351:
307:
278:
111:A general formula
2465:
2442:
2441:
2432:
2423:(510): 499–500,
2412:
2410:
2409:
2404:
2402:
2401:
2383:
2377:
2376:
2341:
2332:
2326:
2325:
2298:
2296:
2295:
2290:
2270:
2225:
2223:
2222:
2217:
2212:
2211:
2199:
2197:
2183:
2129:
2127:
2126:
2121:
2116:
2115:
2103:
2102:
2090:
2089:
2073:
2057:
2055:
2054:
2049:
2044:
2043:
2002:
1999:
1997:
1996:
1958:
1950:
1937:
1935:
1934:
1929:
1927:
1923:
1841:
1839:
1838:
1833:
1828:
1827:
1759:
1757:
1756:
1751:
1749:
1745:
1741:
1732:
1718:
1713:
1701:
1699:
1698:
1693:
1577:
1575:
1574:
1569:
1564:
1563:
1521:
1518:
1515:
1514:
1387:
1385:
1384:
1379:
1374:
1370:
1352:
1344:
1336:
1335:
1288:
1280:
1243:is nonzero, and
1227:
1225:
1224:
1219:
1211:
1203:
1195:
1194:
1159:
1151:
1094:is nonzero, and
1090:, in which case
1082:is nonzero, and
1060:
1058:
1057:
1052:
1050:
1037:
1036:
905:
903:
880:
872:
868:
867:
830:
829:
710:
709:
692:
690:
689:
677:
669:
665:
664:
657:
656:
632:
631:
551:
550:
517:
516:
499:
497:
496:
484:
475:
474:
448:
446:
445:
440:
435:
431:
413:
405:
400:
399:
352:
344:
321:
319:
318:
313:
308:
294:
279:
274:
176:
174:
173:
168:
163:
162:
2473:
2472:
2468:
2467:
2466:
2464:
2463:
2462:
2448:
2447:
2446:
2445:
2393:
2388:
2387:
2385:
2384:
2380:
2339:
2334:
2333:
2329:
2275:
2274:
2272:
2271:
2267:
2262:
2206:
2205:
2200:
2187:
2179:
2178:
2173:
2163:
2151:
2150:
2135:
2107:
2094:
2081:
2076:
2075:
2059:
2038:
2037:
2029:
2023:
2022:
2017:
2004:
2000: and
1991:
1990:
1982:
1976:
1975:
1970:
1960:
1944:
1943:
1920:
1919:
1914:
1908:
1907:
1902:
1890:
1885:
1884:
1882:identity matrix
1874:
1872:Identity matrix
1822:
1821:
1816:
1810:
1809:
1804:
1794:
1782:
1781:
1766:
1764:Diagonal matrix
1729:
1725:
1704:
1703:
1663:
1662:
1651:
1613:, meaning that
1603:
1598:
1558:
1557:
1552:
1546:
1545:
1540:
1530:
1509:
1508:
1503:
1497:
1496:
1491:
1481:
1469:
1468:
1357:
1353:
1330:
1329:
1318:
1312:
1311:
1306:
1290:
1265:
1264:
1189:
1188:
1183:
1177:
1176:
1171:
1161:
1136:
1135:
1120:
1075:is negative.
1048:
1047:
1031:
1030:
1001:
971:
970:
941:
907:
884:
870:
869:
862:
861:
821:
810:
780:
779:
750:
701:
694:
681:
667:
666:
659:
658:
648:
623:
612:
582:
581:
552:
542:
508:
501:
488:
476:
466:
457:
456:
418:
414:
394:
393:
382:
376:
375:
370:
354:
332:
331:
259:
258:
157:
156:
151:
145:
144:
139:
129:
117:
116:
113:
98:
42:stands for the
23:is another 2×2
12:
11:
5:
2471:
2469:
2461:
2460:
2450:
2449:
2444:
2443:
2400:
2396:
2378:
2352:(2): 118–129,
2327:
2309:(4): 222–224,
2288:
2285:
2282:
2264:
2263:
2261:
2258:
2215:
2210:
2204:
2201:
2196:
2193:
2190:
2186:
2181:
2180:
2177:
2174:
2172:
2169:
2168:
2166:
2161:
2158:
2134:
2131:
2119:
2114:
2110:
2106:
2101:
2097:
2093:
2088:
2084:
2047:
2042:
2036:
2033:
2030:
2028:
2025:
2024:
2021:
2018:
2016:
2013:
2010:
2009:
2007:
1995:
1989:
1986:
1983:
1981:
1978:
1977:
1974:
1971:
1969:
1966:
1965:
1963:
1956:
1953:
1926:
1918:
1915:
1913:
1910:
1909:
1906:
1903:
1901:
1898:
1897:
1893:
1873:
1870:
1831:
1826:
1820:
1817:
1815:
1812:
1811:
1808:
1805:
1803:
1800:
1799:
1797:
1792:
1789:
1765:
1762:
1748:
1744:
1738:
1735:
1728:
1724:
1721:
1716:
1711:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1653:If the matrix
1650:
1647:
1602:
1599:
1597:
1594:
1567:
1562:
1556:
1553:
1551:
1548:
1547:
1544:
1541:
1539:
1536:
1535:
1533:
1528:
1525:
1513:
1507:
1504:
1502:
1499:
1498:
1495:
1492:
1490:
1487:
1486:
1484:
1479:
1476:
1412: − 2
1400: − 2
1377:
1373:
1369:
1366:
1363:
1360:
1356:
1350:
1347:
1342:
1339:
1334:
1328:
1325:
1322:
1319:
1317:
1314:
1313:
1310:
1307:
1305:
1302:
1299:
1296:
1295:
1293:
1286:
1283:
1278:
1275:
1272:
1217:
1214:
1209:
1206:
1201:
1198:
1193:
1187:
1184:
1182:
1179:
1178:
1175:
1172:
1170:
1167:
1166:
1164:
1157:
1154:
1149:
1146:
1143:
1119:
1116:
1046:
1043:
1040:
1035:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
912:
910:
902:
899:
896:
893:
890:
887:
883:
878:
875:
873:
871:
866:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
828:
824:
820:
817:
814:
811:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
708:
704:
700:
699:
697:
688:
684:
680:
675:
672:
670:
668:
663:
655:
651:
647:
644:
641:
638:
635:
630:
626:
622:
619:
616:
613:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
549:
545:
541:
538:
535:
532:
529:
526:
523:
520:
515:
511:
507:
506:
504:
495:
491:
487:
482:
479:
477:
473:
469:
465:
464:
438:
434:
430:
427:
424:
421:
417:
411:
408:
403:
398:
392:
389:
386:
383:
381:
378:
377:
374:
371:
369:
366:
363:
360:
359:
357:
350:
347:
342:
339:
311:
306:
303:
300:
297:
292:
289:
286:
282:
277:
272:
269:
266:
166:
161:
155:
152:
150:
147:
146:
143:
140:
138:
135:
134:
132:
127:
124:
112:
109:
44:matrix product
13:
10:
9:
6:
4:
3:
2:
2470:
2459:
2456:
2455:
2453:
2440:
2436:
2431:
2426:
2422:
2418:
2417:
2398:
2394:
2382:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2347:
2346:
2338:
2331:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2303:
2286:
2283:
2280:
2269:
2266:
2259:
2257:
2255:
2251:
2248:is zero, but
2247:
2243:
2239:
2235:
2231:
2226:
2213:
2208:
2202:
2194:
2191:
2188:
2184:
2175:
2170:
2164:
2159:
2156:
2148:
2144:
2141:is zero, but
2140:
2132:
2130:
2117:
2112:
2108:
2104:
2099:
2095:
2091:
2086:
2082:
2071:
2067:
2063:
2045:
2040:
2034:
2031:
2026:
2019:
2014:
2011:
2005:
1993:
1987:
1984:
1979:
1972:
1967:
1961:
1954:
1951:
1941:
1924:
1916:
1911:
1904:
1899:
1891:
1883:
1879:
1871:
1869:
1867:
1863:
1859:
1855:
1851:
1847:
1842:
1829:
1824:
1818:
1813:
1806:
1801:
1795:
1790:
1787:
1779:
1775:
1771:
1763:
1761:
1746:
1742:
1736:
1733:
1726:
1722:
1719:
1714:
1709:
1686:
1680:
1677:
1674:
1671:
1668:
1660:
1656:
1648:
1646:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1600:
1595:
1593:
1591:
1587:
1583:
1578:
1565:
1560:
1554:
1549:
1542:
1537:
1531:
1526:
1523:
1511:
1505:
1500:
1493:
1488:
1482:
1477:
1474:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1417:
1415:
1411:
1407:
1404:nonzero, and
1403:
1399:
1395:
1391:
1375:
1371:
1367:
1364:
1361:
1358:
1354:
1348:
1345:
1340:
1337:
1332:
1326:
1323:
1320:
1315:
1308:
1303:
1300:
1297:
1291:
1284:
1281:
1276:
1273:
1270:
1262:
1258:
1254:
1250:
1246:
1242:
1237:
1235:
1231:
1215:
1212:
1207:
1204:
1199:
1196:
1191:
1185:
1180:
1173:
1168:
1162:
1155:
1152:
1147:
1144:
1141:
1133:
1129:
1125:
1117:
1115:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1076:
1074:
1070:
1066:
1061:
1044:
1041:
1038:
1033:
1024:
1021:
1018:
1015:
1012:
1009:
1003:
995:
992:
989:
986:
983:
980:
974:
964:
961:
958:
955:
952:
949:
943:
935:
932:
929:
926:
923:
920:
914:
908:
900:
897:
894:
891:
888:
885:
881:
876:
874:
864:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
826:
822:
818:
815:
812:
807:
804:
801:
798:
795:
792:
789:
786:
783:
776:
773:
770:
767:
764:
761:
758:
755:
752:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
706:
702:
695:
686:
682:
678:
673:
671:
661:
653:
649:
645:
642:
639:
636:
633:
628:
624:
620:
617:
614:
609:
606:
603:
600:
597:
594:
591:
588:
585:
578:
575:
572:
569:
566:
563:
560:
557:
554:
547:
543:
539:
536:
533:
530:
527:
524:
521:
518:
513:
509:
502:
493:
489:
485:
480:
478:
471:
467:
454:
449:
436:
432:
428:
425:
422:
419:
415:
409:
406:
401:
396:
390:
387:
384:
379:
372:
367:
364:
361:
355:
348:
345:
340:
337:
329:
325:
309:
304:
301:
298:
295:
290:
287:
284:
280:
275:
270:
267:
264:
256:
252:
248:
245:be such that
244:
240:
236:
233:be such that
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
164:
159:
153:
148:
141:
136:
130:
125:
122:
110:
108:
106:
102:
96:
92:
88:
85:) = (−1)(−1)(
84:
80:
76:
72:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
29:
26:
22:
19:
2420:
2414:
2381:
2349:
2343:
2330:
2306:
2300:
2268:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2227:
2146:
2142:
2138:
2136:
2069:
2065:
2061:
1875:
1865:
1861:
1857:
1853:
1849:
1845:
1843:
1777:
1773:
1769:
1767:
1658:
1654:
1652:
1642:
1638:
1634:
1633:= 1, giving
1630:
1626:
1618:
1614:
1606:
1604:
1589:
1585:
1581:
1579:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1418:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1260:
1256:
1248:
1244:
1240:
1238:
1233:
1229:
1131:
1127:
1123:
1121:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1077:
1072:
1068:
1064:
1062:
452:
450:
327:
323:
257:. That is,
254:
250:
246:
242:
238:
234:
230:
222:
218:
214:
210:
202:
198:
194:
190:
186:
182:
178:
114:
94:
90:
86:
82:
78:
74:
70:
66:
62:
60:
55:
47:
39:
35:
31:
27:
20:
17:
15:
1878:eigenvalues
1396:that makes
1253:eigenvalues
1134:. Namely,
227:determinant
101:eigenvalues
2260:References
1880:, the 2×2
1063:Note that
77:, since (−
30:such that
2299:matrix",
2284:×
2032:±
2012:±
1985:−
1940:symmetric
1723:
1710:±
1681:
1341:±
1277:±
1200:±
1148:±
853:−
742:−
322:Then, if
296:τ
291:±
276:δ
271:±
2458:Matrices
2452:Category
1629:= 0 and
69:, then −
38:, where
2439:3621289
2374:1573734
2366:3219099
2323:2689616
2240:= 0 or
229:. Let
225:be its
205:be the
2437:
2372:
2364:
2321:
2058:where
1852:, and
1844:where
1609:is an
1580:where
1435:, and
1388:where
1228:where
241:, and
213:, and
189:, and
177:where
25:matrix
2435:JSTOR
2362:JSTOR
2340:(PDF)
2319:JSTOR
1637:and −
207:trace
2252:and
2145:and
1864:and
1856:= ±√
1848:= ±√
1623:rank
1584:and
1423:and
1110:and
455:is
89:) =
2425:doi
2413:",
2354:doi
2311:doi
2236:or
2137:If
1768:If
1720:exp
1678:exp
1605:If
1519:and
1439:= −
1431:= −
1247:= 4
1086:≠ 4
330:is
209:of
81:)(−
46:of
2454::
2433:,
2421:87
2419:,
2370:MR
2368:,
2360:,
2350:77
2348:,
2342:,
2317:,
2307:53
2305:,
2232:=
2068:,
2064:,
1776:=
1661:,
1645:.
1617:=
1615:MM
1459:=
1455:=
1451:=
1441:BC
1416:.
1236:.
1114:.
255:2s
253:+
249:=
237:=
223:BC
221:−
219:AD
217:=
201:+
197:=
185:,
181:,
93:=
87:RR
34:=
16:A
2427::
2399:2
2395:I
2356::
2313::
2287:2
2281:2
2254:D
2250:A
2246:C
2242:D
2238:A
2234:D
2230:A
2214:.
2209:)
2203:d
2195:d
2192:+
2189:a
2185:C
2176:0
2171:a
2165:(
2160:=
2157:R
2147:D
2143:A
2139:B
2118:.
2113:2
2109:t
2105:=
2100:2
2096:s
2092:+
2087:2
2083:r
2072:)
2070:t
2066:s
2062:r
2060:(
2046:,
2041:)
2035:1
2027:0
2020:0
2015:1
2006:(
1994:)
1988:s
1980:r
1973:r
1968:s
1962:(
1955:t
1952:1
1925:)
1917:1
1912:0
1905:0
1900:1
1892:(
1866:D
1862:A
1858:D
1854:d
1850:A
1846:a
1830:,
1825:)
1819:d
1814:0
1807:0
1802:a
1796:(
1791:=
1788:R
1778:C
1774:B
1770:M
1747:)
1743:A
1737:2
1734:1
1727:(
1715:r
1690:)
1687:A
1684:(
1675:r
1672:=
1669:M
1659:A
1655:M
1643:M
1639:M
1635:M
1631:τ
1627:s
1619:M
1607:M
1590:M
1586:c
1582:b
1566:,
1561:)
1555:0
1550:0
1543:b
1538:0
1532:(
1527:=
1524:R
1512:)
1506:0
1501:c
1494:0
1489:0
1483:(
1478:=
1475:R
1465:M
1461:D
1457:C
1453:B
1449:A
1445:M
1437:A
1433:A
1429:D
1425:τ
1421:δ
1414:s
1410:τ
1406:t
1402:s
1398:τ
1394:δ
1390:s
1376:,
1372:)
1368:I
1365:s
1362:+
1359:M
1355:(
1349:t
1346:1
1338:=
1333:)
1327:s
1324:+
1321:D
1316:C
1309:B
1304:s
1301:+
1298:A
1292:(
1285:t
1282:1
1274:=
1271:R
1261:t
1257:s
1249:δ
1245:τ
1241:δ
1234:τ
1230:t
1216:,
1213:M
1208:t
1205:1
1197:=
1192:)
1186:D
1181:C
1174:B
1169:A
1163:(
1156:t
1153:1
1145:=
1142:R
1132:t
1128:τ
1124:δ
1112:t
1108:s
1104:R
1100:s
1096:t
1092:s
1088:δ
1084:τ
1080:δ
1073:δ
1069:M
1065:R
1045:.
1042:M
1039:=
1034:)
1028:)
1025:s
1022:2
1019:+
1016:D
1013:+
1010:A
1007:(
1004:D
999:)
996:s
993:2
990:+
987:D
984:+
981:A
978:(
975:C
968:)
965:s
962:2
959:+
956:D
953:+
950:A
947:(
944:B
939:)
936:s
933:2
930:+
927:D
924:+
921:A
918:(
915:A
909:(
901:s
898:2
895:+
892:D
889:+
886:A
882:1
877:=
865:)
859:C
856:B
850:D
847:A
844:+
841:D
838:s
835:2
832:+
827:2
823:D
819:+
816:C
813:B
808:C
805:s
802:2
799:+
796:D
793:C
790:+
787:C
784:A
777:B
774:s
771:2
768:+
765:D
762:B
759:+
756:B
753:A
748:C
745:B
739:D
736:A
733:+
730:A
727:s
724:2
721:+
718:C
715:B
712:+
707:2
703:A
696:(
687:2
683:t
679:1
674:=
662:)
654:2
650:s
646:+
643:D
640:s
637:2
634:+
629:2
625:D
621:+
618:B
615:C
610:C
607:s
604:2
601:+
598:C
595:D
592:+
589:A
586:C
579:B
576:s
573:2
570:+
567:D
564:B
561:+
558:B
555:A
548:2
544:s
540:+
537:A
534:s
531:2
528:+
525:C
522:B
519:+
514:2
510:A
503:(
494:2
490:t
486:1
481:=
472:2
468:R
453:R
437:.
433:)
429:I
426:s
423:+
420:M
416:(
410:t
407:1
402:=
397:)
391:s
388:+
385:D
380:C
373:B
368:s
365:+
362:A
356:(
349:t
346:1
341:=
338:R
328:M
324:t
310:.
305:s
302:2
299:+
288:=
285:t
281:,
268:=
265:s
251:τ
247:t
243:t
239:δ
235:s
231:s
215:δ
211:M
203:D
199:A
195:τ
191:D
187:C
183:B
179:A
165:,
160:)
154:D
149:C
142:B
137:A
131:(
126:=
123:M
97:.
95:M
91:R
83:R
79:R
75:M
71:R
67:M
63:R
56:R
48:R
40:R
36:R
32:M
28:R
21:M
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