restart:

with(LinearAlgebra):

with(OreModules):

march('open',"/Users/aquadrat/Documents/Travail/maple/Personnel/RankFactorizationProblem.mla"):

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShsaWJuYW1lR0YoNiVRUi9Vc2Vycy9hcXVhZHJhdC9Eb2N1bWVudHMvVHJhdmFpbC9tYXBsZS9QZXJzb25uZWxGKFFWL0xpYnJhcnkvRnJhbWV3b3Jrcy9NYXBsZS5mcmFtZXdvcmsvVmVyc2lvbnMvMjAyMy9saWJGKFFpby9Vc2Vycy9hcXVhZHJhdC9Eb2N1bWVudHMvVHJhdmFpbC9tYXBsZS9QZXJzb25uZWwvUmFua0ZhY3Rvcml6YXRpb25Qcm9ibGVtLm1sYUYoNyVGL0YwRjE=

with(RankFactorizationProblem);

NzZJLUFudGlEaWFnb25hbEc2IkkwQ2VudHJvSGVybWl0aWFuR0YkSS5GYWN0b3JpemF0aW9uR0YkSTVGaW5pdGVGcmVlUmVzb2x1dGlvbkdGJEktRml0dGluZ0lkZWFsR0YkSTJJc0NlbnRyb0hlcm1pdGlhbkdGJEktSXNJbnZlcnRpYmxlR0YkSSxJc05pbHBvdGVudEdGJEkrSXNTb2x1dGlvbkdGJEkqTGVlTWF0cml4R0YkSSlMZWZ0TGlmdEdGJEklTGlmdEdGJEkyTG9jYWxTeXp5Z3lNb2R1bGVHRiRJMlJhbmtGYWN0b3JpemF0aW9uR0YkSTBSZWR1Y2VkU3l6eWdpZXNHRiRJKlJpZ2h0TGlmdEdGJEkrU2F0dXJhdGlvbkdGJEkvU2ltcGxpZmljYXRpb25HRiRJKlNvbHV0aW9uc0dGJEkpU3l6eWdpZXNHRiQ=

Antidiagonal matrices

AntiDiagonal(1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Xyd6JFs/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyM3Iy1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMUYoSSdNYXRyaXhHRiU3Iy1GPDYjL0kkJWlkR0YoRi8=

AntiDiagonal(2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1NzdRWz86KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRigtRjY2I1EiMUYoNyRGPEY1SSdNYXRyaXhHRiU3Iy1GQDYjL0kkJWlkR0YoRi8=

AntiDiagonal(3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Y1xQWz86KVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNS1GNjYjUSIxRig3JUY1RjxGNTclRjxGNUY1SSdNYXRyaXhHRiU3Iy1GQTYjL0kkJWlkR0YoRi8=

AntiDiagonal(4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1d3VQWz86KVskKm8kLUknTUFUUklYR0YoNiM3JjcmLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIwRihGNUY1LUY2NiNRIjFGKDcmRjVGNUY8RjU3JkY1RjxGNUY1NyZGPEY1RjVGNUknTWF0cml4R0YlNyMtRkI2Iy9JJCVpZEdGKEYv

Lee's matrices

LeeMatrix(2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1V21PWz86KVskKm8kLUknTUFUUklYR0YoNiM3JDckLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjY2I1EtJkltYWdpbmFyeUk7Rig3JEY1LUY2NiNRNiZ1bWludXMwOyZJbWFnaW5hcnlJO0YoSSdNYXRyaXhHRiU3Iy1GQzYjL0kkJWlkR0YoRi8=

MatrixInverse(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1VyVvJFs/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyQ3JC1JJm1mcmFjRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21uR0Y3NiNRIjFGKC1GPDYjUSIyRihGNTckLUklbXJvd0dGNzYkLUkjbW9HRjc2I1EqJnVtaW51czA7RigtRjY2JC1GPDYjUS0mSW1hZ2luYXJ5STtGKEY/RkpJJ01hdHJpeEdGJTcjLUZPNiMvSSQlaWRHRihGLw==

LeeMatrix(3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1R1tPWz86KVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIxRigtRjY2I1EiMEYoLUY2NiNRLSZJbWFnaW5hcnlJO0YoNyVGPEY1Rjw3JUY1RjwtRjY2I1E2JnVtaW51czA7JkltYWdpbmFyeUk7RihJJ01hdHJpeEdGJTcjLUZHNiMvSSQlaWRHRihGLw==

MatrixInverse(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1XyFmJFs/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyU3JS1JJm1mcmFjRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21uR0Y3NiNRIjFGKC1GPDYjUSIyRigtRjw2I1EiMEYoRjU3JUZCRjtGQjclLUklbXJvd0dGNzYkLUkjbW9HRjc2I1EqJnVtaW51czA7RigtRjY2JC1GPDYjUS0mSW1hZ2luYXJ5STtGKEY/RkJGTkknTWF0cml4R0YlNyMtRlM2Iy9JJCVpZEdGKEYv

L := LeeMatrix(4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJMR0YoLUknUlRBQkxFR0YoNiUiNU9bTls/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyY3Ji1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMUYoLUY4NiNRIjBGKC1GODYjUS0mSW1hZ2luYXJ5STtGKEY+NyZGPkY3Rj5GQTcmRj5GN0Y+LUY4NiNRNiZ1bWludXMwOyZJbWFnaW5hcnlJO0YoNyZGN0Y+RkZGPkknTWF0cml4R0YlNyMtRko2Iy9JJCVpZEdGKEYx

MatrixInverse(L);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Z0VOWz86KVskKm8kLUknTUFUUklYR0YoNiM3JjcmLUkmbWZyYWNHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkjbW5HRjc2I1EiMUYoLUY8NiNRIjJGKC1GPDYjUSIwRihGQkY1NyZGQkY1RjVGQjcmLUklbXJvd0dGNzYkLUkjbW9HRjc2I1EqJnVtaW51czA7RigtRjY2JC1GPDYjUS0mSW1hZ2luYXJ5STtGKEY/RkJGQkZONyZGQkZHRk5GQkknTWF0cml4R0YlNyMtRlQ2Iy9JJCVpZEdGKEYv

Test whether or not L is centrohermitian

IsCentroHermitian(L);

SSZmYWxzZUclKnByb3RlY3RlZEc=

Compute the centrohermitian matrix associated with L

H := CentroHermitian(L);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJIR0YoLUknUlRBQkxFR0YoNiUiNVtQTVs/OilbJCpvJC1JJ01BVFJJWEdGKDYjNyY3Ji1JJm1mcmFjRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21uR0Y5NiNRIjFGKC1GPjYjUSIyRigtRjg2JC1GPjYjUS0mSW1hZ2luYXJ5STtGKEZBRkRGNzcmRkRGN0Y3RkQ3Ji1JJW1yb3dHRjk2JC1JI21vR0Y5NiNRKiZ1bWludXMwO0YoRkRGN0Y3Rks3JkY3RktGS0Y3SSdNYXRyaXhHRiU3Iy1GUzYjL0kkJWlkR0YoRjE=

IsCentroHermitian(H);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

We can check that L is not unitary

simplify(Transpose(conjugate(L)).L);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1T09MWz86KVskKm8kLUknTUFUUklYR0YoNiM3JjcmLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSIyRigtRjY2I1EiMEYoRjxGPDcmRjxGNUY8Rjw3JkY8RjxGNUY8NyZGPEY8RjxGNUknTWF0cml4R0YlNyMtRkI2Iy9JJCVpZEdGKEYv

Unitary Lee's matrices

M := LeeMatrix(4,"unitary");

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoLUknUlRBQkxFR0YoNiUiNVsyVSJIXyIpWyQqbyQtSSdNQVRSSVhHRig2IzcmNyYtSSZtZnJhY0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSZtc3FydEdGOTYjLUkjbW5HRjk2I1EiMkYoRkAtRkE2I1EiMEYoLUklbXJvd0dGOTYlLUY4NiQtRkE2I1EtJkltYWdpbmFyeUk7RihGQC1JI21vR0Y5NiNRMSZJbnZpc2libGVUaW1lcztGKEY9RkQ3JkZERjdGREZHNyZGREY3RkQtRkg2JS1GSDYkLUZQNiNRKiZ1bWludXMwO0YoRkpGT0Y9NyZGN0ZERlVGREknTWF0cml4R0YlNyMtRmduNiMvSSQlaWRHRihGMQ==

simplify(Transpose(conjugate(M)).M);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1Jyo0VCJIXyIpWyQqbyQtSSdNQVRSSVhHRig2IzcmNyYtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GNjYjUSIwRihGPEY8NyZGPEY1RjxGPDcmRjxGPEY1Rjw3JkY8RjxGPEY1SSdNYXRyaXhHRiU3Iy1GQjYjL0kkJWlkR0YoRi8=

Unitary Lee's matrices with a parameter

u := 'u';

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ1R0YoRi03I0Yt

R := LeeMatrix(4,"unitary_symbolic",u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoNyVYLCUhR0YoRihbZ2whIiUhISEjMSIlIiUtSSZtZnJhY0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiQtSSNtbkdGMzYjUSIxRigtSSNtaUdGMzYjUSJ1RigtRjg2I1EiMEYoRj9GMUY/RjFGMUY/LUYyNiQtRjg2I1EtJkltYWdpbmFyeUk7RihGO0Y/Rj8tRjI2JC1GODYjUTYmdW1pbnVzMDsmSW1hZ2luYXJ5STtGKEY7Rj9GQkZHRj9JInVHRigsJiokRkwiIiMiIiIhIiNGUDcjNyUtSSdNYXRyaXhHRiU2Iy9JJCVpZEdGKCI1ZzlOWz86KVskKm8kRkxGTQ==

Lee's bijective transformation sending a centrohermitian M to a real matrix M_rho (square case)

M := Matrix([[9+18*I,-225,9+198*I],[0,0,0],[9-198*I,-225,9-18*I]]);

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

IsCentroHermitian(M);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

U := LeeMatrix(3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJVR0YoLUknUlRBQkxFR0YoNiUiNSM0PGNSXyIpWyQqbyQtSSdNQVRSSVhHRig2IzclNyUtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GODYjUSIwRigtRjg2I1EtJkltYWdpbmFyeUk7Rig3JUY+RjdGPjclRjdGPi1GODYjUTYmdW1pbnVzMDsmSW1hZ2luYXJ5STtGKEknTWF0cml4R0YlNyMtRkk2Iy9JJCVpZEdGKEYx

U_inv := MatrixInverse(U);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZVX2ludkdGKC1JJ1JUQUJMRUdGKDYlIjVLKD5jUl8iKVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkmbWZyYWNHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkjbW5HRjk2I1EiMUYoLUY+NiNRIjJGKC1GPjYjUSIwRihGNzclRkRGPUZENyUtSSVtcm93R0Y5NiQtSSNtb0dGOTYjUSomdW1pbnVzMDtGKC1GODYkLUY+NiNRLSZJbWFnaW5hcnlJO0YoRkFGREZQSSdNYXRyaXhHRiU3Iy1GVTYjL0kkJWlkR0YoRjE=

M_rho := U_inv.M.U;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZNX3Job0dGKC1JJ1JUQUJMRUdGKDYlIjVjWGgmUl8iKVskKm8kLUknTUFUUklYR0YoNiM3JTclLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSMxOEYoLUY4NiNRLSZ1bWludXMwOzIyNUYoLUY4NiNRJDE4MEYoNyUtRjg2I1EiMEYoRkVGRTclLUY4NiNRJDIxNkYoRkVGRUknTWF0cml4R0YlNyMtRkw2Iy9JJCVpZEdGKEYx

U.M_rho.U_inv;

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

simplify(%-M);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1IXk2Y1JfIilbJCpvJC1JJ01BVFJJWEdGKDYjNyU3JS1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMEYoRjVGNUY0RjRJJ01hdHJpeEdGJTcjLUY8NiMvSSQlaWRHRihGLw==

Lee's bijective transformation sending a centrohermitian M to a real matrix M_rho (rectangular case)

M := Matrix(2, 5, [[-29, 0, -26, -6*I, -56*I], [56*I, 6*I, -26, 0, -29]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoLUknUlRBQkxFR0YoNiUiNWtkZyZSXyIpWyQqbyQtSSdNQVRSSVhHRig2IzckNyctSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRLCZ1bWludXMwOzI5RigtRjg2I1EiMEYoLUY4NiNRLCZ1bWludXMwOzI2RigtSSVtcm93R0Y5NiQtSSNtb0dGOTYjUSomdW1pbnVzMDtGKC1GRTYlLUY4NiNRIjZGKC1GSDYjUTEmSW52aXNpYmxlVGltZXM7RigtRjg2I1EtJkltYWdpbmFyeUk7RigtRkU2JEZHLUZFNiUtRjg2I1EjNTZGKEZQRlM3J0ZYRktGQUY+RjdJJ01hdHJpeEdGJTcjLUZobjYjL0kkJWlkR0YoRjE=

IsCentroHermitian(M);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

U := LeeMatrix(2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJVR0YoLUknUlRBQkxFR0YoNiUiNV9pZiZSXyIpWyQqbyQtSSdNQVRSSVhHRig2IzckNyQtSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GODYjUS0mSW1hZ2luYXJ5STtGKDckRjctRjg2I1E2JnVtaW51czA7JkltYWdpbmFyeUk7RihJJ01hdHJpeEdGJTcjLUZFNiMvSSQlaWRHRihGMQ==

U_inv := MatrixInverse(U);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZVX2ludkdGKC1JJ1JUQUJMRUdGKDYlIjUjSClmJlJfIilbJCpvJC1JJ01BVFJJWEdGKDYjNyQ3JC1JJm1mcmFjRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JI21uR0Y5NiNRIjFGKC1GPjYjUSIyRihGNzckLUklbXJvd0dGOTYkLUkjbW9HRjk2I1EqJnVtaW51czA7RigtRjg2JC1GPjYjUS0mSW1hZ2luYXJ5STtGKEZBRkxJJ01hdHJpeEdGJTcjLUZRNiMvSSQlaWRHRihGMQ==

V := LeeMatrix(5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJWR0YoLUknUlRBQkxFR0YoNiUiNWNYZiZSXyIpWyQqbyQtSSdNQVRSSVhHRig2IzcnNyctSSNtbkc2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiNRIjFGKC1GODYjUSIwRihGPi1GODYjUS0mSW1hZ2luYXJ5STtGKEY+NydGPkY3Rj5GPkZBNydGPkY+RjdGPkY+NydGPkY3Rj5GPi1GODYjUTYmdW1pbnVzMDsmSW1hZ2luYXJ5STtGKDcnRjdGPkY+RkdGPkknTWF0cml4R0YlNyMtRks2Iy9JJCVpZEdGKEYx

V_inv := MatrixInverse(V);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZWX2ludkdGKC1JJ1JUQUJMRUdGKDYlIjVrXmUmUl8iKVskKm8kLUknTUFUUklYR0YoNiM3JzcnLUkmbWZyYWNHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkjbW5HRjk2I1EiMUYoLUY+NiNRIjJGKC1GPjYjUSIwRihGREZERjc3J0ZERjdGREY3RkQ3J0ZERkRGPUZERkQ3Jy1JJW1yb3dHRjk2JC1JI21vR0Y5NiNRKiZ1bWludXMwO0YoLUY4NiQtRj42I1EtJkltYWdpbmFyeUk7RihGQUZERkRGREZRNydGREZKRkRGUUZESSdNYXRyaXhHRiU3Iy1GVzYjL0kkJWlkR0YoRjE=

M_rho := U_inv.M.V;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZNX3Job0dGKC1JJ1JUQUJMRUdGKDYlIjVHP2UmUl8iKVskKm8kLUknTUFUUklYR0YoNiM3JDcnLUkjbW5HNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSwmdW1pbnVzMDsyOUYoLUY4NiNRIjBGKC1GODYjUSwmdW1pbnVzMDsyNkYoLUY4NiNRLCZ1bWludXMwOzU2RigtRjg2I1ErJnVtaW51czA7NkYoNydGREZHRj5GN0Y+SSdNYXRyaXhHRiU3Iy1GSzYjL0kkJWlkR0YoRjE=

U.M_rho.V_inv;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1c3BkJlJfIilbJCpvJC1JJ01BVFJJWEdGKDYjNyQ3Jy1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EsJnVtaW51czA7MjlGKC1GNjYjUSIwRigtRjY2I1EsJnVtaW51czA7MjZGKC1JJW1yb3dHRjc2JC1JI21vR0Y3NiNRKiZ1bWludXMwO0YoLUZDNiUtRjY2I1EiNkYoLUZGNiNRMSZJbnZpc2libGVUaW1lcztGKC1GNjYjUS0mSW1hZ2luYXJ5STtGKC1GQzYkRkUtRkM2JS1GNjYjUSM1NkYoRk5GUTcnRlZGSUY/RjxGNUknTWF0cml4R0YlNyMtRmZuNiMvSSQlaWRHRihGLw==

simplify(%-M);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSI1OzhkJlJfIilbJCpvJC1JJ01BVFJJWEdGKDYjNyQ3Jy1JI21uRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2I1EiMEYoRjVGNUY1RjVGNEknTWF0cml4R0YlNyMtRjw2Iy9JJCVpZEdGKEYv

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv