How can we generate all n by n matrices with entries {0,1,2,...,m-1} for any natural numbers m and n?

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Hassan Cheraghpour
Hassan Cheraghpour el 10 de En. de 2023
Comentada: Hassan Cheraghpour el 10 de En. de 2023
Dear all,
We know that the number of all n by n matrices with entries {0,1,2,...,m-1} is m^n^2.
I want to generate all of them.
Thank you in advance.
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Dyuman Joshi
Dyuman Joshi el 10 de En. de 2023
Unless m is equal to n^2, the matrix containing 0:m-1 can not be of size n x n.
It is not clear as to what you want to achieve. Please give an example for some values of m and n.
Hassan Cheraghpour
Hassan Cheraghpour el 10 de En. de 2023
Let m=7, and n=2.
I want to generate all 2 by 2 matrices with entries {0,1,2,3,4,5,6}.
We know the number of them is 7^4=2401.
I want to generate them all.

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John D'Errico
John D'Errico el 10 de En. de 2023
Editada: John D'Errico el 10 de En. de 2023
Ran in:
So, for the simpler case of how to solve for ALL 2x2 matrices, containing the integers 0:6, just work in base 7.
M = reshape(dec2base(0:7^4-1,7)' - '0',[2,2,7^4])
M =
M(:,:,1) = 0 0 0 0 M(:,:,2) = 0 0 0 1 M(:,:,3) = 0 0 0 2 M(:,:,4) = 0 0 0 3 M(:,:,5) = 0 0 0 4 M(:,:,6) = 0 0 0 5 M(:,:,7) = 0 0 0 6 M(:,:,8) = 0 1 0 0 M(:,:,9) = 0 1 0 1 M(:,:,10) = 0 1 0 2 M(:,:,11) = 0 1 0 3 M(:,:,12) = 0 1 0 4 M(:,:,13) = 0 1 0 5 M(:,:,14) = 0 1 0 6 M(:,:,15) = 0 2 0 0 M(:,:,16) = 0 2 0 1 M(:,:,17) = 0 2 0 2 M(:,:,18) = 0 2 0 3 M(:,:,19) = 0 2 0 4 M(:,:,20) = 0 2 0 5 M(:,:,21) = 0 2 0 6 M(:,:,22) = 0 3 0 0 M(:,:,23) = 0 3 0 1 M(:,:,24) = 0 3 0 2 M(:,:,25) = 0 3 0 3 M(:,:,26) = 0 3 0 4 M(:,:,27) = 0 3 0 5 M(:,:,28) = 0 3 0 6 M(:,:,29) = 0 4 0 0 M(:,:,30) = 0 4 0 1 M(:,:,31) = 0 4 0 2 M(:,:,32) = 0 4 0 3 M(:,:,33) = 0 4 0 4 M(:,:,34) = 0 4 0 5 M(:,:,35) = 0 4 0 6 M(:,:,36) = 0 5 0 0 M(:,:,37) = 0 5 0 1 M(:,:,38) = 0 5 0 2 M(:,:,39) = 0 5 0 3 M(:,:,40) = 0 5 0 4 M(:,:,41) = 0 5 0 5 M(:,:,42) = 0 5 0 6 M(:,:,43) = 0 6 0 0 M(:,:,44) = 0 6 0 1 M(:,:,45) = 0 6 0 2 M(:,:,46) = 0 6 0 3 M(:,:,47) = 0 6 0 4 M(:,:,48) = 0 6 0 5 M(:,:,49) = 0 6 0 6 M(:,:,50) = 0 0 1 0 M(:,:,51) = 0 0 1 1 M(:,:,52) = 0 0 1 2 M(:,:,53) = 0 0 1 3 M(:,:,54) = 0 0 1 4 M(:,:,55) = 0 0 1 5 M(:,:,56) = 0 0 1 6 M(:,:,57) = 0 1 1 0 M(:,:,58) = 0 1 1 1 M(:,:,59) = 0 1 1 2 M(:,:,60) = 0 1 1 3 M(:,:,61) = 0 1 1 4 M(:,:,62) = 0 1 1 5 M(:,:,63) = 0 1 1 6 M(:,:,64) = 0 2 1 0 M(:,:,65) = 0 2 1 1 M(:,:,66) = 0 2 1 2 M(:,:,67) = 0 2 1 3 M(:,:,68) = 0 2 1 4 M(:,:,69) = 0 2 1 5 M(:,:,70) = 0 2 1 6 M(:,:,71) = 0 3 1 0 M(:,:,72) = 0 3 1 1 M(:,:,73) = 0 3 1 2 M(:,:,74) = 0 3 1 3 M(:,:,75) = 0 3 1 4 M(:,:,76) = 0 3 1 5 M(:,:,77) = 0 3 1 6 M(:,:,78) = 0 4 1 0 M(:,:,79) = 0 4 1 1 M(:,:,80) = 0 4 1 2 M(:,:,81) = 0 4 1 3 M(:,:,82) = 0 4 1 4 M(:,:,83) = 0 4 1 5 M(:,:,84) = 0 4 1 6 M(:,:,85) = 0 5 1 0 M(:,:,86) = 0 5 1 1 M(:,:,87) = 0 5 1 2 M(:,:,88) = 0 5 1 3 M(:,:,89) = 0 5 1 4 M(:,:,90) = 0 5 1 5 M(:,:,91) = 0 5 1 6 M(:,:,92) = 0 6 1 0 M(:,:,93) = 0 6 1 1 M(:,:,94) = 0 6 1 2 M(:,:,95) = 0 6 1 3 M(:,:,96) = 0 6 1 4 M(:,:,97) = 0 6 1 5 M(:,:,98) = 0 6 1 6 M(:,:,99) = 0 0 2 0 M(:,:,100) = 0 0 2 1 M(:,:,101) = 0 0 2 2 M(:,:,102) = 0 0 2 3 M(:,:,103) = 0 0 2 4 M(:,:,104) = 0 0 2 5 M(:,:,105) = 0 0 2 6 M(:,:,106) = 0 1 2 0 M(:,:,107) = 0 1 2 1 M(:,:,108) = 0 1 2 2 M(:,:,109) = 0 1 2 3 M(:,:,110) = 0 1 2 4 M(:,:,111) = 0 1 2 5 M(:,:,112) = 0 1 2 6 M(:,:,113) = 0 2 2 0 M(:,:,114) = 0 2 2 1 M(:,:,115) = 0 2 2 2 M(:,:,116) = 0 2 2 3 M(:,:,117) = 0 2 2 4 M(:,:,118) = 0 2 2 5 M(:,:,119) = 0 2 2 6 M(:,:,120) = 0 3 2 0 M(:,:,121) = 0 3 2 1 M(:,:,122) = 0 3 2 2 M(:,:,123) = 0 3 2 3 M(:,:,124) = 0 3 2 4 M(:,:,125) = 0 3 2 5 M(:,:,126) = 0 3 2 6 M(:,:,127) = 0 4 2 0 M(:,:,128) = 0 4 2 1 M(:,:,129) = 0 4 2 2 M(:,:,130) = 0 4 2 3 M(:,:,131) = 0 4 2 4 M(:,:,132) = 0 4 2 5 M(:,:,133) = 0 4 2 6 M(:,:,134) = 0 5 2 0 M(:,:,135) = 0 5 2 1 M(:,:,136) = 0 5 2 2 M(:,:,137) = 0 5 2 3 M(:,:,138) = 0 5 2 4 M(:,:,139) = 0 5 2 5 M(:,:,140) = 0 5 2 6 M(:,:,141) = 0 6 2 0 M(:,:,142) = 0 6 2 1 M(:,:,143) = 0 6 2 2 M(:,:,144) = 0 6 2 3 M(:,:,145) = 0 6 2 4 M(:,:,146) = 0 6 2 5 M(:,:,147) = 0 6 2 6 M(:,:,148) = 0 0 3 0 M(:,:,149) = 0 0 3 1 M(:,:,150) = 0 0 3 2 M(:,:,151) = 0 0 3 3 M(:,:,152) = 0 0 3 4 M(:,:,153) = 0 0 3 5 M(:,:,154) = 0 0 3 6 M(:,:,155) = 0 1 3 0 M(:,:,156) = 0 1 3 1 M(:,:,157) = 0 1 3 2 M(:,:,158) = 0 1 3 3 M(:,:,159) = 0 1 3 4 M(:,:,160) = 0 1 3 5 M(:,:,161) = 0 1 3 6 M(:,:,162) = 0 2 3 0 M(:,:,163) = 0 2 3 1 M(:,:,164) = 0 2 3 2 M(:,:,165) = 0 2 3 3 M(:,:,166) = 0 2 3 4 M(:,:,167) = 0 2 3 5 M(:,:,168) = 0 2 3 6 M(:,:,169) = 0 3 3 0 M(:,:,170) = 0 3 3 1 M(:,:,171) = 0 3 3 2 M(:,:,172) = 0 3 3 3 M(:,:,173) = 0 3 3 4 M(:,:,174) = 0 3 3 5 M(:,:,175) = 0 3 3 6 M(:,:,176) = 0 4 3 0 M(:,:,177) = 0 4 3 1 M(:,:,178) = 0 4 3 2 M(:,:,179) = 0 4 3 3 M(:,:,180) = 0 4 3 4 M(:,:,181) = 0 4 3 5 M(:,:,182) = 0 4 3 6 M(:,:,183) = 0 5 3 0 M(:,:,184) = 0 5 3 1 M(:,:,185) = 0 5 3 2 M(:,:,186) = 0 5 3 3 M(:,:,187) = 0 5 3 4 M(:,:,188) = 0 5 3 5 M(:,:,189) = 0 5 3 6 M(:,:,190) = 0 6 3 0 M(:,:,191) = 0 6 3 1 M(:,:,192) = 0 6 3 2 M(:,:,193) = 0 6 3 3 M(:,:,194) = 0 6 3 4 M(:,:,195) = 0 6 3 5 M(:,:,196) = 0 6 3 6 M(:,:,197) = 0 0 4 0 M(:,:,198) = 0 0 4 1 M(:,:,199) = 0 0 4 2 M(:,:,200) = 0 0 4 3 M(:,:,201) = 0 0 4 4 M(:,:,202) = 0 0 4 5 M(:,:,203) = 0 0 4 6 M(:,:,204) = 0 1 4 0 M(:,:,205) = 0 1 4 1 M(:,:,206) = 0 1 4 2 M(:,:,207) = 0 1 4 3 M(:,:,208) = 0 1 4 4 M(:,:,209) = 0 1 4 5 M(:,:,210) = 0 1 4 6 M(:,:,211) = 0 2 4 0 M(:,:,212) = 0 2 4 1 M(:,:,213) = 0 2 4 2 M(:,:,214) = 0 2 4 3 M(:,:,215) = 0 2 4 4 M(:,:,216) = 0 2 4 5 M(:,:,217) = 0 2 4 6 M(:,:,218) = 0 3 4 0 M(:,:,219) = 0 3 4 1 M(:,:,220) = 0 3 4 2 M(:,:,221) = 0 3 4 3 M(:,:,222) = 0 3 4 4 M(:,:,223) = 0 3 4 5 M(:,:,224) = 0 3 4 6 M(:,:,225) = 0 4 4 0 M(:,:,226) = 0 4 4 1 M(:,:,227) = 0 4 4 2 M(:,:,228) = 0 4 4 3 M(:,:,229) = 0 4 4 4 M(:,:,230) = 0 4 4 5 M(:,:,231) = 0 4 4 6 M(:,:,232) = 0 5 4 0 M(:,:,233) = 0 5 4 1 M(:,:,234) = 0 5 4 2 M(:,:,235) = 0 5 4 3 M(:,:,236) = 0 5 4 4 M(:,:,237) = 0 5 4 5 M(:,:,238) = 0 5 4 6 M(:,:,239) = 0 6 4 0 M(:,:,240) = 0 6 4 1 M(:,:,241) = 0 6 4 2 M(:,:,242) = 0 6 4 3 M(:,:,243) = 0 6 4 4 M(:,:,244) = 0 6 4 5 M(:,:,245) = 0 6 4 6 M(:,:,246) = 0 0 5 0 M(:,:,247) = 0 0 5 1 M(:,:,248) = 0 0 5 2 M(:,:,249) = 0 0 5 3 M(:,:,250) = 0 0 5 4 M(:,:,251) = 0 0 5 5 M(:,:,252) = 0 0 5 6 M(:,:,253) = 0 1 5 0 M(:,:,254) = 0 1 5 1 M(:,:,255) = 0 1 5 2 M(:,:,256) = 0 1 5 3 M(:,:,257) = 0 1 5 4 M(:,:,258) = 0 1 5 5 M(:,:,259) = 0 1 5 6 M(:,:,260) = 0 2 5 0 M(:,:,261) = 0 2 5 1 M(:,:,262) = 0 2 5 2 M(:,:,263) = 0 2 5 3 M(:,:,264) = 0 2 5 4 M(:,:,265) = 0 2 5 5 M(:,:,266) = 0 2 5 6 M(:,:,267) = 0 3 5 0 M(:,:,268) = 0 3 5 1 M(:,:,269) = 0 3 5 2 M(:,:,270) = 0 3 5 3 M(:,:,271) = 0 3 5 4 M(:,:,272) = 0 3 5 5 M(:,:,273) = 0 3 5 6 M(:,:,274) = 0 4 5 0 M(:,:,275) = 0 4 5 1 M(:,:,276) = 0 4 5 2 M(:,:,277) = 0 4 5 3 M(:,:,278) = 0 4 5 4 M(:,:,279) = 0 4 5 5 M(:,:,280) = 0 4 5 6 M(:,:,281) = 0 5 5 0 M(:,:,282) = 0 5 5 1 M(:,:,283) = 0 5 5 2 M(:,:,284) = 0 5 5 3 M(:,:,285) = 0 5 5 4 M(:,:,286) = 0 5 5 5 M(:,:,287) = 0 5 5 6 M(:,:,288) = 0 6 5 0 M(:,:,289) = 0 6 5 1 M(:,:,290) = 0 6 5 2 M(:,:,291) = 0 6 5 3 M(:,:,292) = 0 6 5 4 M(:,:,293) = 0 6 5 5 M(:,:,294) = 0 6 5 6 M(:,:,295) = 0 0 6 0 M(:,:,296) = 0 0 6 1 M(:,:,297) = 0 0 6 2 M(:,:,298) = 0 0 6 3 M(:,:,299) = 0 0 6 4 M(:,:,300) = 0 0 6 5 M(:,:,301) = 0 0 6 6 M(:,:,302) = 0 1 6 0 M(:,:,303) = 0 1 6 1 M(:,:,304) = 0 1 6 2 M(:,:,305) = 0 1 6 3 M(:,:,306) = 0 1 6 4 M(:,:,307) = 0 1 6 5 M(:,:,308) = 0 1 6 6 M(:,:,309) = 0 2 6 0 M(:,:,310) = 0 2 6 1 M(:,:,311) = 0 2 6 2 M(:,:,312) = 0 2 6 3 M(:,:,313) = 0 2 6 4 M(:,:,314) = 0 2 6 5 M(:,:,315) = 0 2 6 6 M(:,:,316) = 0 3 6 0 M(:,:,317) = 0 3 6 1 M(:,:,318) = 0 3 6 2 M(:,:,319) = 0 3 6 3 M(:,:,320) = 0 3 6 4 M(:,:,321) = 0 3 6 5 M(:,:,322) = 0 3 6 6 M(:,:,323) = 0 4 6 0 M(:,:,324) = 0 4 6 1 M(:,:,325) = 0 4 6 2 M(:,:,326) = 0 4 6 3 M(:,:,327) = 0 4 6 4 M(:,:,328) = 0 4 6 5 M(:,:,329) = 0 4 6 6 M(:,:,330) = 0 5 6 0 M(:,:,331) = 0 5 6 1 M(:,:,332) = 0 5 6 2 M(:,:,333) = 0 5 6 3 M(:,:,334) = 0 5 6 4 M(:,:,335) = 0 5 6 5 M(:,:,336) = 0 5 6 6 M(:,:,337) = 0 6 6 0 M(:,:,338) = 0 6 6 1 M(:,:,339) = 0 6 6 2 M(:,:,340) = 0 6 6 3 M(:,:,341) = 0 6 6 4 M(:,:,342) = 0 6 6 5 M(:,:,343) = 0 6 6 6 M(:,:,344) = 1 0 0 0 M(:,:,345) = 1 0 0 1 M(:,:,346) = 1 0 0 2 M(:,:,347) = 1 0 0 3 M(:,:,348) = 1 0 0 4 M(:,:,349) = 1 0 0 5 M(:,:,350) = 1 0 0 6 M(:,:,351) = 1 1 0 0 M(:,:,352) = 1 1 0 1 M(:,:,353) = 1 1 0 2 M(:,:,354) = 1 1 0 3 M(:,:,355) = 1 1 0 4 M(:,:,356) = 1 1 0 5 M(:,:,357) = 1 1 0 6 M(:,:,358) = 1 2 0 0 M(:,:,359) = 1 2 0 1 M(:,:,360) = 1 2 0 2 M(:,:,361) = 1 2 0 3 M(:,:,362) = 1 2 0 4 M(:,:,363) = 1 2 0 5 M(:,:,364) = 1 2 0 6 M(:,:,365) = 1 3 0 0 M(:,:,366) = 1 3 0 1 M(:,:,367) = 1 3 0 2 M(:,:,368) = 1 3 0 3 M(:,:,369) = 1 3 0 4 M(:,:,370) = 1 3 0 5 M(:,:,371) = 1 3 0 6 M(:,:,372) = 1 4 0 0 M(:,:,373) = 1 4 0 1 M(:,:,374) = 1 4 0 2 M(:,:,375) = 1 4 0 3 M(:,:,376) = 1 4 0 4 M(:,:,377) = 1 4 0 5 M(:,:,378) = 1 4 0 6 M(:,:,379) = 1 5 0 0 M(:,:,380) = 1 5 0 1 M(:,:,381) = 1 5 0 2 M(:,:,382) = 1 5 0 3 M(:,:,383) = 1 5 0 4 M(:,:,384) = 1 5 0 5 M(:,:,385) = 1 5 0 6 M(:,:,386) = 1 6 0 0 M(:,:,387) = 1 6 0 1 M(:,:,388) = 1 6 0 2 M(:,:,389) = 1 6 0 3 M(:,:,390) = 1 6 0 4 M(:,:,391) = 1 6 0 5 M(:,:,392) = 1 6 0 6 M(:,:,393) = 1 0 1 0 M(:,:,394) = 1 0 1 1 M(:,:,395) = 1 0 1 2 M(:,:,396) = 1 0 1 3 M(:,:,397) = 1 0 1 4 M(:,:,398) = 1 0 1 5 M(:,:,399) = 1 0 1 6 M(:,:,400) = 1 1 1 0 M(:,:,401) = 1 1 1 1 M(:,:,402) = 1 1 1 2 M(:,:,403) = 1 1 1 3 M(:,:,404) = 1 1 1 4 M(:,:,405) = 1 1 1 5 M(:,:,406) = 1 1 1 6 M(:,:,407) = 1 2 1 0 M(:,:,408) = 1 2 1 1 M(:,:,409) = 1 2 1 2 M(:,:,410) = 1 2 1 3 M(:,:,411) = 1 2 1 4 M(:,:,412) = 1 2 1 5 M(:,:,413) = 1 2 1 6 M(:,:,414) = 1 3 1 0 M(:,:,415) = 1 3 1 1 M(:,:,416) = 1 3 1 2 M(:,:,417) = 1 3 1 3 M(:,:,418) = 1 3 1 4 M(:,:,419) = 1 3 1 5 M(:,:,420) = 1 3 1 6 M(:,:,421) = 1 4 1 0 M(:,:,422) = 1 4 1 1 M(:,:,423) = 1 4 1 2 M(:,:,424) = 1 4 1 3 M(:,:,425) = 1 4 1 4 M(:,:,426) = 1 4 1 5 M(:,:,427) = 1 4 1 6 M(:,:,428) = 1 5 1 0 M(:,:,429) = 1 5 1 1 M(:,:,430) = 1 5 1 2 M(:,:,431) = 1 5 1 3 M(:,:,432) = 1 5 1 4 M(:,:,433) = 1 5 1 5 M(:,:,434) = 1 5 1 6 M(:,:,435) = 1 6 1 0 M(:,:,436) = 1 6 1 1 M(:,:,437) = 1 6 1 2 M(:,:,438) = 1 6 1 3 M(:,:,439) = 1 6 1 4 M(:,:,440) = 1 6 1 5 M(:,:,441) = 1 6 1 6 M(:,:,442) = 1 0 2 0 M(:,:,443) = 1 0 2 1 M(:,:,444) = 1 0 2 2 M(:,:,445) = 1 0 2 3 M(:,:,446) = 1 0 2 4 M(:,:,447) = 1 0 2 5 M(:,:,448) = 1 0 2 6 M(:,:,449) = 1 1 2 0 M(:,:,450) = 1 1 2 1 M(:,:,451) = 1 1 2 2 M(:,:,452) = 1 1 2 3 M(:,:,453) = 1 1 2 4 M(:,:,454) = 1 1 2 5 M(:,:,455) = 1 1 2 6 M(:,:,456) = 1 2 2 0 M(:,:,457) = 1 2 2 1 M(:,:,458) = 1 2 2 2 M(:,:,459) = 1 2 2 3 M(:,:,460) = 1 2 2 4 M(:,:,461) = 1 2 2 5 M(:,:,462) = 1 2 2 6 M(:,:,463) = 1 3 2 0 M(:,:,464) = 1 3 2 1 M(:,:,465) = 1 3 2 2 M(:,:,466) = 1 3 2 3 M(:,:,467) = 1 3 2 4 M(:,:,468) = 1 3 2 5 M(:,:,469) = 1 3 2 6 M(:,:,470) = 1 4 2 0 M(:,:,471) = 1 4 2 1 M(:,:,472) = 1 4 2 2 M(:,:,473) = 1 4 2 3 M(:,:,474) = 1 4 2 4 M(:,:,475) = 1 4 2 5 M(:,:,476) = 1 4 2 6 M(:,:,477) = 1 5 2 0 M(:,:,478) = 1 5 2 1 M(:,:,479) = 1 5 2 2 M(:,:,480) = 1 5 2 3 M(:,:,481) = 1 5 2 4 M(:,:,482) = 1 5 2 5 M(:,:,483) = 1 5 2 6 M(:,:,484) = 1 6 2 0 M(:,:,485) = 1 6 2 1 M(:,:,486) = 1 6 2 2 M(:,:,487) = 1 6 2 3 M(:,:,488) = 1 6 2 4 M(:,:,489) = 1 6 2 5 M(:,:,490) = 1 6 2 6 M(:,:,491) = 1 0 3 0 M(:,:,492) = 1 0 3 1 M(:,:,493) = 1 0 3 2 M(:,:,494) = 1 0 3 3 M(:,:,495) = 1 0 3 4 M(:,:,496) = 1 0 3 5 M(:,:,497) = 1 0 3 6 M(:,:,498) = 1 1 3 0 M(:,:,499) = 1 1 3 1 M(:,:,500) = 1 1 3 2 M(:,:,501) = 1 1 3 3 M(:,:,502) = 1 1 3 4 M(:,:,503) = 1 1 3 5 M(:,:,504) = 1 1 3 6 M(:,:,505) = 1 2 3 0 M(:,:,506) = 1 2 3 1 M(:,:,507) = 1 2 3 2 M(:,:,508) = 1 2 3 3 M(:,:,509) = 1 2 3 4 M(:,:,510) = 1 2 3 5 M(:,:,511) = 1 2 3 6 M(:,:,512) = 1 3 3 0 M(:,:,513) = 1 3 3 1 M(:,:,514) = 1 3 3 2 M(:,:,515) = 1 3 3 3 M(:,:,516) = 1 3 3 4 M(:,:,517) = 1 3 3 5 M(:,:,518) = 1 3 3 6 M(:,:,519) = 1 4 3 0 M(:,:,520) = 1 4 3 1 M(:,:,521) = 1 4 3 2 M(:,:,522) = 1 4 3 3 M(:,:,523) = 1 4 3 4 M(:,:,524) = 1 4 3 5 M(:,:,525) = 1 4 3 6 M(:,:,526) = 1 5 3 0 M(:,:,527) = 1 5 3 1 M(:,:,528) = 1 5 3 2 M(:,:,529) = 1 5 3 3 M(:,:,530) = 1 5 3 4 M(:,:,531) = 1 5 3 5 M(:,:,532) = 1 5 3 6 M(:,:,533) = 1 6 3 0 M(:,:,534) = 1 6 3 1 M(:,:,535) = 1 6 3 2 M(:,:,536) = 1 6 3 3 M(:,:,537) = 1 6 3 4 M(:,:,538) = 1 6 3 5 M(:,:,539) = 1 6 3 6 M(:,:,540) = 1 0 4 0 M(:,:,541) = 1 0 4 1 M(:,:,542) = 1 0 4 2 M(:,:,543) = 1 0 4 3 M(:,:,544) = 1 0 4 4 M(:,:,545) = 1 0 4 5 M(:,:,546) = 1 0 4 6 M(:,:,547) = 1 1 4 0 M(:,:,548) = 1 1 4 1 M(:,:,549) = 1 1 4 2 M(:,:,550) = 1 1 4 3 M(:,:,551) = 1 1 4 4 M(:,:,552) = 1 1 4 5 M(:,:,553) = 1 1 4 6 M(:,:,554) = 1 2 4 0 M(:,:,555) = 1 2 4 1 M(:,:,556) = 1 2 4 2 M(:,:,557) = 1 2 4 3 M(:,:,558) = 1 2 4 4 M(:,:,559) = 1 2 4 5 M(:,:,560) = 1 2 4 6 M(:,:,561) = 1 3 4 0 M(:,:,562) = 1 3 4 1 M(:,:,563) = 1 3 4 2 M(:,:,564) = 1 3 4 3 M(:,:,565) = 1 3 4 4 M(:,:,566) = 1 3 4 5 M(:,:,567) = 1 3 4 6 M(:,:,568) = 1 4 4 0 M(:,:,569) = 1 4 4 1 M(:,:,570) = 1 4 4 2 M(:,:,571) = 1 4 4 3 M(:,:,572) = 1 4 4 4 M(:,:,573) = 1 4 4 5 M(:,:,574) = 1 4 4 6 M(:,:,575) = 1 5 4 0 M(:,:,576) = 1 5 4 1 M(:,:,577) = 1 5 4 2 M(:,:,578) = 1 5 4 3 M(:,:,579) = 1 5 4 4 M(:,:,580) = 1 5 4 5 M(:,:,581) = 1 5 4 6 M(:,:,582) = 1 6 4 0 M(:,:,583) = 1 6 4 1 M(:,:,584) = 1 6 4 2 M(:,:,585) = 1 6 4 3 M(:,:,586) = 1 6 4 4 M(:,:,587) = 1 6 4 5 M(:,:,588) = 1 6 4 6 M(:,:,589) = 1 0 5 0 M(:,:,590) = 1 0 5 1 M(:,:,591) = 1 0 5 2 M(:,:,592) = 1 0 5 3 M(:,:,593) = 1 0 5 4 M(:,:,594) = 1 0 5 5 M(:,:,595) = 1 0 5 6 M(:,:,596) = 1 1 5 0 M(:,:,597) = 1 1 5 1 M(:,:,598) = 1 1 5 2 M(:,:,599) = 1 1 5 3 M(:,:,600) = 1 1 5 4 M(:,:,601) = 1 1 5 5 M(:,:,602) = 1 1 5 6 M(:,:,603) = 1 2 5 0 M(:,:,604) = 1 2 5 1 M(:,:,605) = 1 2 5 2 M(:,:,606) = 1 2 5 3 M(:,:,607) = 1 2 5 4 M(:,:,608) = 1 2 5 5 M(:,:,609) = 1 2 5 6 M(:,:,610) = 1 3 5 0 M(:,:,611) = 1 3 5 1 M(:,:,612) = 1 3 5 2 M(:,:,613) = 1 3 5 3 M(:,:,614) = 1 3 5 4 M(:,:,615) = 1 3 5 5 M(:,:,616) = 1 3 5 6 M(:,:,617) = 1 4 5 0 M(:,:,618) = 1 4 5 1 M(:,:,619) = 1 4 5 2 M(:,:,620) = 1 4 5 3 M(:,:,621) = 1 4 5 4 M(:,:,622) = 1 4 5 5 M(:,:,623) = 1 4 5 6 M(:,:,624) = 1 5 5 0 M(:,:,625) = 1 5 5 1 M(:,:,626) = 1 5 5 2 M(:,:,627) = 1 5 5 3 M(:,:,628) = 1 5 5 4 M(:,:,629) = 1 5 5 5 M(:,:,630) = 1 5 5 6 M(:,:,631) = 1 6 5 0 M(:,:,632) = 1 6 5 1 M(:,:,633) = 1 6 5 2 M(:,:,634) = 1 6 5 3 M(:,:,635) = 1 6 5 4 M(:,:,636) = 1 6 5 5 M(:,:,637) = 1 6 5 6 M(:,:,638) = 1 0 6 0 M(:,:,639) = 1 0 6 1 M(:,:,640) = 1 0 6 2 M(:,:,641) = 1 0 6 3 M(:,:,642) = 1 0 6 4 M(:,:,643) = 1 0 6 5 M(:,:,644) = 1 0 6 6 M(:,:,645) = 1 1 6 0 M(:,:,646) = 1 1 6 1 M(:,:,647) = 1 1 6 2 M(:,:,648) = 1 1 6 3 M(:,:,649) = 1 1 6 4 M(:,:,650) = 1 1 6 5 M(:,:,651) = 1 1 6 6 M(:,:,652) = 1 2 6 0 M(:,:,653) = 1 2 6 1 M(:,:,654) = 1 2 6 2 M(:,:,655) = 1 2 6 3 M(:,:,656) = 1 2 6 4 M(:,:,657) = 1 2 6 5 M(:,:,658) = 1 2 6 6 M(:,:,659) = 1 3 6 0 M(:,:,660) = 1 3 6 1 M(:,:,661) = 1 3 6 2 M(:,:,662) = 1 3 6 3 M(:,:,663) = 1 3 6 4 M(:,:,664) = 1 3 6 5 M(:,:,665) = 1 3 6 6 M(:,:,666) = 1 4 6 0 M(:,:,667) = 1 4 6 1 M(:,:,668) = 1 4 6 2 M(:,:,669) = 1 4 6 3 M(:,:,670) = 1 4 6 4 M(:,:,671) = 1 4 6 5 M(:,:,672) = 1 4 6 6 M(:,:,673) = 1 5 6 0 M(:,:,674) = 1 5 6 1 M(:,:,675) = 1 5 6 2 M(:,:,676) = 1 5 6 3 M(:,:,677) = 1 5 6 4 M(:,:,678) = 1 5 6 5 M(:,:,679) = 1 5 6 6 M(:,:,680) = 1 6 6 0 M(:,:,681) = 1 6 6 1 M(:,:,682) = 1 6 6 2 M(:,:,683) = 1 6 6 3 M(:,:,684) = 1 6 6 4 M(:,:,685) = 1 6 6 5 M(:,:,686) = 1 6 6 6 M(:,:,687) = 2 0 0 0 M(:,:,688) = 2 0 0 1 M(:,:,689) = 2 0 0 2 M(:,:,690) = 2 0 0 3 M(:,:,691) = 2 0 0 4 M(:,:,692) = 2 0 0 5 M(:,:,693) = 2 0 0 6 M(:,:,694) = 2 1 0 0 M(:,:,695) = 2 1 0 1 M(:,:,696) = 2 1 0 2 M(:,:,697) = 2 1 0 3 M(:,:,698) = 2 1 0 4 M(:,:,699) = 2 1 0 5 M(:,:,700) = 2 1 0 6 M(:,:,701) = 2 2 0 0 M(:,:,702) = 2 2 0 1 M(:,:,703) = 2 2 0 2 M(:,:,704) = 2 2 0 3 M(:,:,705) = 2 2 0 4 M(:,:,706) = 2 2 0 5 M(:,:,707) = 2 2 0 6 M(:,:,708) = 2 3 0 0 M(:,:,709) = 2 3 0 1 M(:,:,710) = 2 3 0 2 M(:,:,711) = 2 3 0 3 M(:,:,712) = 2 3 0 4 M(:,:,713) = 2 3 0 5 M(:,:,714) = 2 3 0 6 M(:,:,715) = 2 4 0 0 M(:,:,716) = 2 4 0 1 M(:,:,717) = 2 4 0 2 M(:,:,718) = 2 4 0 3 M(:,:,719) = 2 4 0 4 M(:,:,720) = 2 4 0 5 M(:,:,721) = 2 4 0 6 M(:,:,722) = 2 5 0 0 M(:,:,723) = 2 5 0 1 M(:,:,724) = 2 5 0 2 M(:,:,725) = 2 5 0 3 M(:,:,726) = 2 5 0 4 M(:,:,727) = 2 5 0 5 M(:,:,728) = 2 5 0 6 M(:,:,729) = 2 6 0 0 M(:,:,730) = 2 6 0 1 M(:,:,731) = 2 6 0 2 M(:,:,732) = 2 6 0 3 M(:,:,733) = 2 6 0 4 M(:,:,734) = 2 6 0 5 M(:,:,735) = 2 6 0 6 M(:,:,736) = 2 0 1 0 M(:,:,737) = 2 0 1 1 M(:,:,738) = 2 0 1 2 M(:,:,739) = 2 0 1 3 M(:,:,740) = 2 0 1 4 M(:,:,741) = 2 0 1 5 M(:,:,742) = 2 0 1 6 M(:,:,743) = 2 1 1 0 M(:,:,744) = 2 1 1 1 M(:,:,745) = 2 1 1 2 M(:,:,746) = 2 1 1 3 M(:,:,747) = 2 1 1 4 M(:,:,748) = 2 1 1 5 M(:,:,749) = 2 1 1 6 M(:,:,750) = 2 2 1 0 M(:,:,751) = 2 2 1 1 M(:,:,752) = 2 2 1 2 M(:,:,753) = 2 2 1 3 M(:,:,754) = 2 2 1 4 M(:,:,755) = 2 2 1 5 M(:,:,756) = 2 2 1 6 M(:,:,757) = 2 3 1 0 M(:,:,758) = 2 3 1 1 M(:,:,759) = 2 3 1 2 M(:,:,760) = 2 3 1 3 M(:,:,761) = 2 3 1 4 M(:,:,762) = 2 3 1 5 M(:,:,763) = 2 3 1 6 M(:,:,764) = 2 4 1 0 M(:,:,765) = 2 4 1 1 M(:,:,766) = 2 4 1 2 M(:,:,767) = 2 4 1 3 M(:,:,768) = 2 4 1 4 M(:,:,769) = 2 4 1 5 M(:,:,770) = 2 4 1 6 M(:,:,771) = 2 5 1 0 M(:,:,772) = 2 5 1 1 M(:,:,773) = 2 5 1 2 M(:,:,774) = 2 5 1 3 M(:,:,775) = 2 5 1 4 M(:,:,776) = 2 5 1 5 M(:,:,777) = 2 5 1 6 M(:,:,778) = 2 6 1 0 M(:,:,779) = 2 6 1 1 M(:,:,780) = 2 6 1 2 M(:,:,781) = 2 6 1 3 M(:,:,782) = 2 6 1 4 M(:,:,783) = 2 6 1 5 M(:,:,784) = 2 6 1 6 M(:,:,785) = 2 0 2 0 M(:,:,786) = 2 0 2 1 M(:,:,787) = 2 0 2 2 M(:,:,788) = 2 0 2 3 M(:,:,789) = 2 0 2 4 M(:,:,790) = 2 0 2 5 M(:,:,791) = 2 0 2 6 M(:,:,792) = 2 1 2 0 M(:,:,793) = 2 1 2 1 M(:,:,794) = 2 1 2 2 M(:,:,795) = 2 1 2 3 M(:,:,796) = 2 1 2 4 M(:,:,797) = 2 1 2 5 M(:,:,798) = 2 1 2 6 M(:,:,799) = 2 2 2 0 M(:,:,800) = 2 2 2 1 M(:,:,801) = 2 2 2 2 M(:,:,802) = 2 2 2 3 M(:,:,803) = 2 2 2 4 M(:,:,804) = 2 2 2 5 M(:,:,805) = 2 2 2 6 M(:,:,806) = 2 3 2 0 M(:,:,807) = 2 3 2 1 M(:,:,808) = 2 3 2 2 M(:,:,809) = 2 3 2 3 M(:,:,810) = 2 3 2 4 M(:,:,811) = 2 3 2 5 M(:,:,812) = 2 3 2 6 M(:,:,813) = 2 4 2 0 M(:,:,814) = 2 4 2 1 M(:,:,815) = 2 4 2 2 M(:,:,816) = 2 4 2 3 M(:,:,817) = 2 4 2 4 M(:,:,818) = 2 4 2 5 M(:,:,819) = 2 4 2 6 M(:,:,820) = 2 5 2 0 M(:,:,821) = 2 5 2 1 M(:,:,822) = 2 5 2 2 M(:,:,823) = 2 5 2 3 M(:,:,824) = 2 5 2 4 M(:,:,825) = 2 5 2 5 M(:,:,826) = 2 5 2 6 M(:,:,827) = 2 6 2 0 M(:,:,828) = 2 6 2 1 M(:,:,829) = 2 6 2 2 M(:,:,830) = 2 6 2 3 M(:,:,831) = 2 6 2 4 M(:,:,832) = 2 6 2 5 M(:,:,833) = 2 6 2 6 M(:,:,834) = 2 0 3 0 M(:,:,835) = 2 0 3 1 M(:,:,836) = 2 0 3 2 M(:,:,837) = 2 0 3 3 M(:,:,838) = 2 0 3 4 M(:,:,839) = 2 0 3 5 M(:,:,840) = 2 0 3 6 M(:,:,841) = 2 1 3 0 M(:,:,842) = 2 1 3 1 M(:,:,843) = 2 1 3 2 M(:,:,844) = 2 1 3 3 M(:,:,845) = 2 1 3 4 M(:,:,846) = 2 1 3 5 M(:,:,847) = 2 1 3 6 M(:,:,848) = 2 2 3 0 M(:,:,849) = 2 2 3 1 M(:,:,850) = 2 2 3 2 M(:,:,851) = 2 2 3 3 M(:,:,852) = 2 2 3 4 M(:,:,853) = 2 2 3 5 M(:,:,854) = 2 2 3 6 M(:,:,855) = 2 3 3 0 M(:,:,856) = 2 3 3 1 M(:,:,857) = 2 3 3 2 M(:,:,858) = 2 3 3 3 M(:,:,859) = 2 3 3 4 M(:,:,860) = 2 3 3 5 M(:,:,861) = 2 3 3 6 M(:,:,862) = 2 4 3 0 M(:,:,863) = 2 4 3 1 M(:,:,864) = 2 4 3 2 M(:,:,865) = 2 4 3 3 M(:,:,866) = 2 4 3 4 M(:,:,867) = 2 4 3 5 M(:,:,868) = 2 4 3 6 M(:,:,869) = 2 5 3 0 M(:,:,870) = 2 5 3 1 M(:,:,871) = 2 5 3 2 M(:,:,872) = 2 5 3 3 M(:,:,873) = 2 5 3 4 M(:,:,874) = 2 5 3 5 M(:,:,875) = 2 5 3 6 M(:,:,876) = 2 6 3 0 M(:,:,877) = 2 6 3 1 M(:,:,878) = 2 6 3 2 M(:,:,879) = 2 6 3 3 M(:,:,880) = 2 6 3 4 M(:,:,881) = 2 6 3 5 M(:,:,882) = 2 6 3 6 M(:,:,883) = 2 0 4 0 M(:,:,884) = 2 0 4 1 M(:,:,885) = 2 0 4 2 M(:,:,886) = 2 0 4 3 M(:,:,887) = 2 0 4 4 M(:,:,888) = 2 0 4 5 M(:,:,889) = 2 0 4 6 M(:,:,890) = 2 1 4 0 M(:,:,891) = 2 1 4 1 M(:,:,892) = 2 1 4 2 M(:,:,893) = 2 1 4 3 M(:,:,894) = 2 1 4 4 M(:,:,895) = 2 1 4 5 M(:,:,896) = 2 1 4 6 M(:,:,897) = 2 2 4 0 M(:,:,898) = 2 2 4 1 M(:,:,899) = 2 2 4 2 M(:,:,900) = 2 2 4 3 M(:,:,901) = 2 2 4 4 M(:,:,902) = 2 2 4 5 M(:,:,903) = 2 2 4 6 M(:,:,904) = 2 3 4 0 M(:,:,905) = 2 3 4 1 M(:,:,906) = 2 3 4 2 M(:,:,907) = 2 3 4 3 M(:,:,908) = 2 3 4 4 M(:,:,909) = 2 3 4 5 M(:,:,910) = 2 3 4 6 M(:,:,911) = 2 4 4 0 M(:,:,912) = 2 4 4 1 M(:,:,913) = 2 4 4 2 M(:,:,914) = 2 4 4 3 M(:,:,915) = 2 4 4 4 M(:,:,916) = 2 4 4 5 M(:,:,917) = 2 4 4 6 M(:,:,918) = 2 5 4 0 M(:,:,919) = 2 5 4 1 M(:,:,920) = 2 5 4 2 M(:,:,921) = 2 5 4 3 M(:,:,922) = 2 5 4 4 M(:,:,923) = 2 5 4 5 M(:,:,924) = 2 5 4 6 M(:,:,925) = 2 6 4 0 M(:,:,926) = 2 6 4 1 M(:,:,927) = 2 6 4 2 M(:,:,928) = 2 6 4 3 M(:,:,929) = 2 6 4 4 M(:,:,930) = 2 6 4 5 M(:,:,931) = 2 6 4 6 M(:,:,932) = 2 0 5 0 M(:,:,933) = 2 0 5 1 M(:,:,934) = 2 0 5 2 M(:,:,935) = 2 0 5 3 M(:,:,936) = 2 0 5 4 M(:,:,937) = 2 0 5 5 M(:,:,938) = 2 0 5 6 M(:,:,939) = 2 1 5 0 M(:,:,940) = 2 1 5 1 M(:,:,941) = 2 1 5 2 M(:,:,942) = 2 1 5 3 M(:,:,943) = 2 1 5 4 M(:,:,944) = 2 1 5 5 M(:,:,945) = 2 1 5 6 M(:,:,946) = 2 2 5 0 M(:,:,947) = 2 2 5 1 M(:,:,948) = 2 2 5 2 M(:,:,949) = 2 2 5 3 M(:,:,950) = 2 2 5 4 M(:,:,951) = 2 2 5 5 M(:,:,952) = 2 2 5 6 M(:,:,953) = 2 3 5 0 M(:,:,954) = 2 3 5 1 M(:,:,955) = 2 3 5 2 M(:,:,956) = 2 3 5 3 M(:,:,957) = 2 3 5 4 M(:,:,958) = 2 3 5 5 M(:,:,959) = 2 3 5 6 M(:,:,960) = 2 4 5 0 M(:,:,961) = 2 4 5 1 M(:,:,962) = 2 4 5 2 M(:,:,963) = 2 4 5 3 M(:,:,964) = 2 4 5 4 M(:,:,965) = 2 4 5 5 M(:,:,966) = 2 4 5 6 M(:,:,967) = 2 5 5 0 M(:,:,968) = 2 5 5 1 M(:,:,969) = 2 5 5 2 M(:,:,970) = 2 5 5 3 M(:,:,971) = 2 5 5 4 M(:,:,972) = 2 5 5 5 M(:,:,973) = 2 5 5 6 M(:,:,974) = 2 6 5 0 M(:,:,975) = 2 6 5 1 M(:,:,976) = 2 6 5 2 M(:,:,977) = 2 6 5 3 M(:,:,978) = 2 6 5 4 M(:,:,979) = 2 6 5 5 M(:,:,980) = 2 6 5 6 M(:,:,981) = 2 0 6 0 M(:,:,982) = 2 0 6 1 M(:,:,983) = 2 0 6 2 M(:,:,984) = 2 0 6 3 M(:,:,985) = 2 0 6 4 M(:,:,986) = 2 0 6 5 M(:,:,987) = 2 0 6 6 M(:,:,988) = 2 1 6 0 M(:,:,989) = 2 1 6 1 M(:,:,990) = 2 1 6 2 M(:,:,991) = 2 1 6 3 M(:,:,992) = 2 1 6 4 M(:,:,993) = 2 1 6 5 M(:,:,994) = 2 1 6 6 M(:,:,995) = 2 2 6 0 M(:,:,996) = 2 2 6 1 M(:,:,997) = 2 2 6 2 M(:,:,998) = 2 2 6 3 M(:,:,999) = 2 2 6 4 M(:,:,1000) = 2 2 6 5 M(:,:,1001) = 2 2 6 6 M(:,:,1002) = 2 3 6 0 M(:,:,1003) = 2 3 6 1 M(:,:,1004) = 2 3 6 2 M(:,:,1005) = 2 3 6 3 M(:,:,1006) = 2 3 6 4 M(:,:,1007) = 2 3 6 5 M(:,:,1008) = 2 3 6 6 M(:,:,1009) = 2 4 6 0 M(:,:,1010) = 2 4 6 1 M(:,:,1011) = 2 4 6 2 M(:,:,1012) = 2 4 6 3 M(:,:,1013) = 2 4 6 4 M(:,:,1014) = 2 4 6 5 M(:,:,1015) = 2 4 6 6 M(:,:,1016) = 2 5 6 0 M(:,:,1017) = 2 5 6 1 M(:,:,1018) = 2 5 6 2 M(:,:,1019) = 2 5 6 3 M(:,:,1020) = 2 5 6 4 M(:,:,1021) = 2 5 6 5 M(:,:,1022) = 2 5 6 6 M(:,:,1023) = 2 6 6 0 M(:,:,1024) = 2 6 6 1 M(:,:,1025) = 2 6 6 2 M(:,:,1026) = 2 6 6 3 M(:,:,1027) = 2 6 6 4 M(:,:,1028) = 2 6 6 5 M(:,:,1029) = 2 6 6 6 M(:,:,1030) = 3 0 0 0 M(:,:,1031) = 3 0 0 1 M(:,:,1032) = 3 0 0 2 M(:,:,1033) = 3 0 0 3 M(:,:,1034) = 3 0 0 4 M(:,:,1035) = 3 0 0 5 M(:,:,1036) = 3 0 0 6 M(:,:,1037) = 3 1 0 0 M(:,:,1038) = 3 1 0 1 M(:,:,1039) = 3 1 0 2 M(:,:,1040) = 3 1 0 3 M(:,:,1041) = 3 1 0 4 M(:,:,1042) = 3 1 0 5 M(:,:,1043) = 3 1 0 6 M(:,:,1044) = 3 2 0 0 M(:,:,1045) = 3 2 0 1 M(:,:,1046) = 3 2 0 2 M(:,:,1047) = 3 2 0 3 M(:,:,1048) = 3 2 0 4 M(:,:,1049) = 3 2 0 5 M(:,:,1050) = 3 2 0 6 M(:,:,1051) = 3 3 0 0 M(:,:,1052) = 3 3 0 1 M(:,:,1053) = 3 3 0 2 M(:,:,1054) = 3 3 0 3 M(:,:,1055) = 3 3 0 4 M(:,:,1056) = 3 3 0 5 M(:,:,1057) = 3 3 0 6 M(:,:,1058) = 3 4 0 0 M(:,:,1059) = 3 4 0 1 M(:,:,1060) = 3 4 0 2 M(:,:,1061) = 3 4 0 3 M(:,:,1062) = 3 4 0 4 M(:,:,1063) = 3 4 0 5 M(:,:,1064) = 3 4 0 6 M(:,:,1065) = 3 5 0 0 M(:,:,1066) = 3 5 0 1 M(:,:,1067) = 3 5 0 2 M(:,:,1068) = 3 5 0 3 M(:,:,1069) = 3 5 0 4 M(:,:,1070) = 3 5 0 5 M(:,:,1071) = 3 5 0 6 M(:,:,1072) = 3 6 0 0 M(:,:,1073) = 3 6 0 1 M(:,:,1074) = 3 6 0 2 M(:,:,1075) = 3 6 0 3 M(:,:,1076) = 3 6 0 4 M(:,:,1077) = 3 6 0 5 M(:,:,1078) = 3 6 0 6 M(:,:,1079) = 3 0 1 0 M(:,:,1080) = 3 0 1 1 M(:,:,1081) = 3 0 1 2 M(:,:,1082) = 3 0 1 3 M(:,:,1083) = 3 0 1 4 M(:,:,1084) = 3 0 1 5 M(:,:,1085) = 3 0 1 6 M(:,:,1086) = 3 1 1 0 M(:,:,1087) = 3 1 1 1 M(:,:,1088) = 3 1 1 2 M(:,:,1089) = 3 1 1 3 M(:,:,1090) = 3 1 1 4 M(:,:,1091) = 3 1 1 5 M(:,:,1092) = 3 1 1 6 M(:,:,1093) = 3 2 1 0 M(:,:,1094) = 3 2 1 1 M(:,:,1095) = 3 2 1 2 M(:,:,1096) = 3 2 1 3 M(:,:,1097) = 3 2 1 4 M(:,:,1098) = 3 2 1 5 M(:,:,1099) = 3 2 1 6 M(:,:,1100) = 3 3 1 0 M(:,:,1101) = 3 3 1 1 M(:,:,1102) = 3 3 1 2 M(:,:,1103) = 3 3 1 3 M(:,:,1104) = 3 3 1 4 M(:,:,1105) = 3 3 1 5 M(:,:,1106) = 3 3 1 6 M(:,:,1107) = 3 4 1 0 M(:,:,1108) = 3 4 1 1 M(:,:,1109) = 3 4 1 2 M(:,:,1110) = 3 4 1 3 M(:,:,1111) = 3 4 1 4 M(:,:,1112) = 3 4 1 5 M(:,:,1113) = 3 4 1 6 M(:,:,1114) = 3 5 1 0 M(:,:,1115) = 3 5 1 1 M(:,:,1116) = 3 5 1 2 M(:,:,1117) = 3 5 1 3 M(:,:,1118) = 3 5 1 4 M(:,:,1119) = 3 5 1 5 M(:,:,1120) = 3 5 1 6 M(:,:,1121) = 3 6 1 0 M(:,:,1122) = 3 6 1 1 M(:,:,1123) = 3 6 1 2 M(:,:,1124) = 3 6 1 3 M(:,:,1125) = 3 6 1 4 M(:,:,1126) = 3 6 1 5 M(:,:,1127) = 3 6 1 6 M(:,:,1128) = 3 0 2 0 M(:,:,1129) = 3 0 2 1 M(:,:,1130) = 3 0 2 2 M(:,:,1131) = 3 0 2 3 M(:,:,1132) = 3 0 2 4 M(:,:,1133) = 3 0 2 5 M(:,:,1134) = 3 0 2 6 M(:,:,1135) = 3 1 2 0 M(:,:,1136) = 3 1 2 1 M(:,:,1137) = 3 1 2 2 M(:,:,1138) = 3 1 2 3 M(:,:,1139) = 3 1 2 4 M(:,:,1140) = 3 1 2 5 M(:,:,1141) = 3 1 2 6 M(:,:,1142) = 3 2 2 0 M(:,:,1143) = 3 2 2 1 M(:,:,1144) = 3 2 2 2 M(:,:,1145) = 3 2 2 3 M(:,:,1146) = 3 2 2 4 M(:,:,1147) = 3 2 2 5 M(:,:,1148) = 3 2 2 6 M(:,:,1149) = 3 3 2 0 M(:,:,1150) = 3 3 2 1 M(:,:,1151) = 3 3 2 2 M(:,:,1152) = 3 3 2 3 M(:,:,1153) = 3 3 2 4 M(:,:,1154) = 3 3 2 5 M(:,:,1155) = 3 3 2 6 M(:,:,1156) = 3 4 2 0 M(:,:,1157) = 3 4 2 1 M(:,:,1158) = 3 4 2 2 M(:,:,1159) = 3 4 2 3 M(:,:,1160) = 3 4 2 4 M(:,:,1161) = 3 4 2 5 M(:,:,1162) = 3 4 2 6 M(:,:,1163) = 3 5 2 0 M(:,:,1164) = 3 5 2 1 M(:,:,1165) = 3 5 2 2 M(:,:,1166) = 3 5 2 3 M(:,:,1167) = 3 5 2 4 M(:,:,1168) = 3 5 2 5 M(:,:,1169) = 3 5 2 6 M(:,:,1170) = 3 6 2 0 M(:,:,1171) = 3 6 2 1 M(:,:,1172) = 3 6 2 2 M(:,:,1173) = 3 6 2 3 M(:,:,1174) = 3 6 2 4 M(:,:,1175) = 3 6 2 5 M(:,:,1176) = 3 6 2 6 M(:,:,1177) = 3 0 3 0 M(:,:,1178) = 3 0 3 1 M(:,:,1179) = 3 0 3 2 M(:,:,1180) = 3 0 3 3 M(:,:,1181) = 3 0 3 4 M(:,:,1182) = 3 0 3 5 M(:,:,1183) = 3 0 3 6 M(:,:,1184) = 3 1 3 0 M(:,:,1185) = 3 1 3 1 M(:,:,1186) = 3 1 3 2 M(:,:,1187) = 3 1 3 3 M(:,:,1188) = 3 1 3 4 M(:,:,1189) = 3 1 3 5 M(:,:,1190) = 3 1 3 6 M(:,:,1191) = 3 2 3 0 M(:,:,1192) = 3 2 3 1 M(:,:,1193) = 3 2 3 2 M(:,:,1194) = 3 2 3 3 M(:,:,1195) = 3 2 3 4 M(:,:,1196) = 3 2 3 5 M(:,:,1197) = 3 2 3 6 M(:,:,1198) = 3 3 3 0 M(:,:,1199) = 3 3 3 1 M(:,:,1200) = 3 3 3 2 M(:,:,1201) = 3 3 3 3 M(:,:,1202) = 3 3 3 4 M(:,:,1203) = 3 3 3 5 M(:,:,1204) = 3 3 3 6 M(:,:,1205) = 3 4 3 0 M(:,:,1206) = 3 4 3 1 M(:,:,1207) = 3 4 3 2 M(:,:,1208) = 3 4 3 3 M(:,:,1209) = 3 4 3 4 M(:,:,1210) = 3 4 3 5 M(:,:,1211) = 3 4 3 6 M(:,:,1212) = 3 5 3 0 M(:,:,1213) = 3 5 3 1 M(:,:,1214) = 3 5 3 2 M(:,:,1215) = 3 5 3 3 M(:,:,1216) = 3 5 3 4 M(:,:,1217) = 3 5 3 5 M(:,:,1218) = 3 5 3 6 M(:,:,1219) = 3 6 3 0 M(:,:,1220) = 3 6 3 1 M(:,:,1221) = 3 6 3 2 M(:,:,1222) = 3 6 3 3 M(:,:,1223) = 3 6 3 4 M(:,:,1224) = 3 6 3 5 M(:,:,1225) = 3 6 3 6 M(:,:,1226) = 3 0 4 0 M(:,:,1227) = 3 0 4 1 M(:,:,1228) = 3 0 4 2 M(:,:,1229) = 3 0 4 3 M(:,:,1230) = 3 0 4 4 M(:,:,1231) = 3 0 4 5 M(:,:,1232) = 3 0 4 6 M(:,:,1233) = 3 1 4 0 M(:,:,1234) = 3 1 4 1 M(:,:,1235) = 3 1 4 2 M(:,:,1236) = 3 1 4 3 M(:,:,1237) = 3 1 4 4 M(:,:,1238) = 3 1 4 5 M(:,:,1239) = 3 1 4 6 M(:,:,1240) = 3 2 4 0 M(:,:,1241) = 3 2 4 1 M(:,:,1242) = 3 2 4 2 M(:,:,1243) = 3 2 4 3 M(:,:,1244) = 3 2 4 4 M(:,:,1245) = 3 2 4 5 M(:,:,1246) = 3 2 4 6 M(:,:,1247) = 3 3 4 0 M(:,:,1248) = 3 3 4 1 M(:,:,1249) = 3 3 4 2 M(:,:,1250) = 3 3 4 3 M(:,:,1251) = 3 3 4 4 M(:,:,1252) = 3 3 4 5 M(:,:,1253) = 3 3 4 6 M(:,:,1254) = 3 4 4 0 M(:,:,1255) = 3 4 4 1 M(:,:,1256) = 3 4 4 2 M(:,:,1257) = 3 4 4 3 M(:,:,1258) = 3 4 4 4 M(:,:,1259) = 3 4 4 5 M(:,:,1260) = 3 4 4 6 M(:,:,1261) = 3 5 4 0 M(:,:,1262) = 3 5 4 1 M(:,:,1263) = 3 5 4 2 M(:,:,1264) = 3 5 4 3 M(:,:,1265) = 3 5 4 4 M(:,:,1266) = 3 5 4 5 M(:,:,1267) = 3 5 4 6 M(:,:,1268) = 3 6 4 0 M(:,:,1269) = 3 6 4 1 M(:,:,1270) = 3 6 4 2 M(:,:,1271) = 3 6 4 3 M(:,:,1272) = 3 6 4 4 M(:,:,1273) = 3 6 4 5 M(:,:,1274) = 3 6 4 6 M(:,:,1275) = 3 0 5 0 M(:,:,1276) = 3 0 5 1 M(:,:,1277) = 3 0 5 2 M(:,:,1278) = 3 0 5 3 M(:,:,1279) = 3 0 5 4 M(:,:,1280) = 3 0 5 5 M(:,:,1281) = 3 0 5 6 M(:,:,1282) = 3 1 5 0 M(:,:,1283) = 3 1 5 1 M(:,:,1284) = 3 1 5 2 M(:,:,1285) = 3 1 5 3 M(:,:,1286) = 3 1 5 4 M(:,:,1287) = 3 1 5 5 M(:,:,1288) = 3 1 5 6 M(:,:,1289) = 3 2 5 0 M(:,:,1290) = 3 2 5 1 M(:,:,1291) = 3 2 5 2 M(:,:,1292) = 3 2 5 3 M(:,:,1293) = 3 2 5 4 M(:,:,1294) = 3 2 5 5 M(:,:,1295) = 3 2 5 6 M(:,:,1296) = 3 3 5 0 M(:,:,1297) = 3 3 5 1 M(:,:,1298) = 3 3 5 2 M(:,:,1299) = 3 3 5 3 M(:,:,1300) = 3 3 5 4 M(:,:,1301) = 3 3 5 5 M(:,:,1302) = 3 3 5 6 M(:,:,1303) = 3 4 5 0 M(:,:,1304) = 3 4 5 1 M(:,:,1305) = 3 4 5 2 M(:,:,1306) = 3 4 5 3 M(:,:,1307) = 3 4 5 4 M(:,:,1308) = 3 4 5 5 M(:,:,1309) = 3 4 5 6 M(:,:,1310) = 3 5 5 0 M(:,:,1311) = 3 5 5 1 M(:,:,1312) = 3 5 5 2 M(:,:,1313) = 3 5 5 3 M(:,:,1314) = 3 5 5 4 M(:,:,1315) = 3 5 5 5 M(:,:,1316) = 3 5 5 6 M(:,:,1317) = 3 6 5 0 M(:,:,1318) = 3 6 5 1 M(:,:,1319) = 3 6 5 2 M(:,:,1320) = 3 6 5 3 M(:,:,1321) = 3 6 5 4 M(:,:,1322) = 3 6 5 5 M(:,:,1323) = 3 6 5 6 M(:,:,1324) = 3 0 6 0 M(:,:,1325) = 3 0 6 1 M(:,:,1326) = 3 0 6 2 M(:,:,1327) = 3 0 6 3 M(:,:,1328) = 3 0 6 4 M(:,:,1329) = 3 0 6 5 M(:,:,1330) = 3 0 6 6 M(:,:,1331) = 3 1 6 0 M(:,:,1332) = 3 1 6 1 M(:,:,1333) = 3 1 6 2 M(:,:,1334) = 3 1 6 3 M(:,:,1335) = 3 1 6 4 M(:,:,1336) = 3 1 6 5 M(:,:,1337) = 3 1 6 6 M(:,:,1338) = 3 2 6 0 M(:,:,1339) = 3 2 6 1 M(:,:,1340) = 3 2 6 2 M(:,:,1341) = 3 2 6 3 M(:,:,1342) = 3 2 6 4 M(:,:,1343) = 3 2 6 5 M(:,:,1344) = 3 2 6 6 M(:,:,1345) = 3 3 6 0 M(:,:,1346) = 3 3 6 1 M(:,:,1347) = 3 3 6 2 M(:,:,1348) = 3 3 6 3 M(:,:,1349) = 3 3 6 4 M(:,:,1350) = 3 3 6 5 M(:,:,1351) = 3 3 6 6 M(:,:,1352) = 3 4 6 0 M(:,:,1353) = 3 4 6 1 M(:,:,1354) = 3 4 6 2 M(:,:,1355) = 3 4 6 3 M(:,:,1356) = 3 4 6 4 M(:,:,1357) = 3 4 6 5 M(:,:,1358) = 3 4 6 6 M(:,:,1359) = 3 5 6 0 M(:,:,1360) = 3 5 6 1 M(:,:,1361) = 3 5 6 2 M(:,:,1362) = 3 5 6 3 M(:,:,1363) = 3 5 6 4 M(:,:,1364) = 3 5 6 5 M(:,:,1365) = 3 5 6 6 M(:,:,1366) = 3 6 6 0 M(:,:,1367) = 3 6 6 1 M(:,:,1368) = 3 6 6 2 M(:,:,1369) = 3 6 6 3 M(:,:,1370) = 3 6 6 4 M(:,:,1371) = 3 6 6 5 M(:,:,1372) = 3 6 6 6 M(:,:,1373) = 4 0 0 0 M(:,:,1374) = 4 0 0 1 M(:,:,1375) = 4 0 0 2 M(:,:,1376) = 4 0 0 3 M(:,:,1377) = 4 0 0 4 M(:,:,1378) = 4 0 0 5 M(:,:,1379) = 4 0 0 6 M(:,:,1380) = 4 1 0 0 M(:,:,1381) = 4 1 0 1 M(:,:,1382) = 4 1 0 2 M(:,:,1383) = 4 1 0 3 M(:,:,1384) = 4 1 0 4 M(:,:,1385) = 4 1 0 5 M(:,:,1386) = 4 1 0 6 M(:,:,1387) = 4 2 0 0 M(:,:,1388) = 4 2 0 1 M(:,:,1389) = 4 2 0 2 M(:,:,1390) = 4 2 0 3 M(:,:,1391) = 4 2 0 4 M(:,:,1392) = 4 2 0 5 M(:,:,1393) = 4 2 0 6 M(:,:,1394) = 4 3 0 0 M(:,:,1395) = 4 3 0 1 M(:,:,1396) = 4 3 0 2 M(:,:,1397) = 4 3 0 3 M(:,:,1398) = 4 3 0 4 M(:,:,1399) = 4 3 0 5 M(:,:,1400) = 4 3 0 6 M(:,:,1401) = 4 4 0 0 M(:,:,1402) = 4 4 0 1 M(:,:,1403) = 4 4 0 2 M(:,:,1404) = 4 4 0 3 M(:,:,1405) = 4 4 0 4 M(:,:,1406) = 4 4 0 5 M(:,:,1407) = 4 4 0 6 M(:,:,1408) = 4 5 0 0 M(:,:,1409) = 4 5 0 1 M(:,:,1410) = 4 5 0 2 M(:,:,1411) = 4 5 0 3 M(:,:,1412) = 4 5 0 4 M(:,:,1413) = 4 5 0 5 M(:,:,1414) = 4 5 0 6 M(:,:,1415) = 4 6 0 0 M(:,:,1416) = 4 6 0 1 M(:,:,1417) = 4 6 0 2 M(:,:,1418) = 4 6 0 3 M(:,:,1419) = 4 6 0 4 M(:,:,1420) = 4 6 0 5 M(:,:,1421) = 4 6 0 6
This is a 2x2x2401 array, so not terribly large.
whos M
Name Size Bytes Class Attributes M 2x2x2401 76832 double
Access any single one of those arrays by an index into the third dimension of M.
M(:,:,1000)
ans = 2×2
2 2 6 5
M(:,:,2401)
ans = 2×2
6 6 6 6
And as I said, I expect you will soon be asking why when you try to solve a larger problem, that you ran out of memeory, and how to solve that. The answer in advance is to not use brute force, to not solve the problem this way. Even if you wanted to compute the corresponding 3x3 arrays, this will still require close to a gigabyte of RAM to store them all.

Más respuestas (2)

Steven Lord
Steven Lord el 10 de En. de 2023
Ran in:
How large are the values of m and n in which you're interested?
What are you planning to do with these matrices once you've generated them?
If they're even moderately large, trying to brute force whatever calculation you're planning to do on them is likely to take an infeasible amount of time. Let's say m and n are each equal to 10 and you can process a million such matrices per second.
m = 10;
n = 10;
numMatrices = m^(n^2);
format longg
y = years(seconds(numMatrices/1e6))
y =
3.16887385068114e+86
According to the timeline of the far future Earth is long gone, either engulfed by the Sun or impacting on what remains after the Sun has expanded and collapsed.
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Steven Lord
Steven Lord el 10 de En. de 2023
Ran in:
Okay, so you want idempotent matrices over Z_m. If you want m and n both to be 7:
m = 7;
n = 7;
numMatrices = m^(n^2);
format longg
y = years(seconds(numMatrices/1e6))
y =
8.14158406429933e+27
Earth is still gone. But if m is 7 and n is 2:
m = 7;
n = 2;
numMatrices = m^(n^2)
numMatrices =
2401
Now you have something manageable. You don't need to generate all 2401 of those matrices, though. If you use the equations on that Wikipedia page you can iterate over all the values for a and d to give you two potential values of b*c. If a particular (a, d) pair doesn't give the same potential value of b*c, that pair doesn't produce an idempotent matrix. If it does solve the equations for b and c to see if there is a solution.

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John D'Errico
John D'Errico el 10 de En. de 2023
Editada: John D'Errico el 10 de En. de 2023
Ran in:
Assuming you mean with replacement, then yes, the total number of those matrices is m^(n^2). For m and n even reasonably large, this number of matrices is immense. It will require often more memory than you will have on your computer. For example, with m=n=10, the number of such matrices would be 10^100. This would require only
seven trigintillion, four hundred fifty novemvigintillion, five hundred eighty octovigintillion, five hundred ninety six septenvigintillion, nine hundred twenty three sexvigintillion, eight hundred twenty eight quinvigintillion, one hundred twenty five quattuorvigintillion
gigabytes of RAM to store them all.
Ok, maybe that is a bit large. It is a number that is more then the number of elementary particles in the universe (which by some estimates would be roughly 10^80.) Instead, with m=n=5, that should surely be doable, right?
5^25*25*8/2^30
ans = 5.5511e+10
So to store those small arrays, now you need only 55 billion gigabytes of RAM. Not even 55 gigabytes, but 55 billion GIGABYTES.
Do you understand just how big this problem gets, and how rapidly it grows?
The point is, whenever someone wants to do something like this, you need to understand you are almost certainly trying to solve a problem using brute force, when that is just a flat out bad idea. Instead learn to use tools of optimization, of dynamic programming, etc.
  2 comentarios
John D'Errico
John D'Errico el 10 de En. de 2023
I showed that for m=n=5 this is still wildly impossible. And you only want 7.
Your other comments suggest that n==2 though. So all 2x2 matrices containing the numbers 0:6, with replacement.
Knowing that as soon as you learn to solve that, your problem will expand, and then you will be asking why you ran out of memory... this is just a bad idea. Again, you would be far better off using other tools and avoiding brute force solutions. I'll post the answer separately though.

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