How can calculate the matrix inverse

How can calculate R in the following equation in MATLAB

C=D*R*D' % C, D, D' are square matrices, D' is the transpose of D

Answer 1

[Walter Roberson]

C=D*R*D'

implies that

C * inv(D') = D * R
inv(D) * C * inv(D') = R

so

R = D\C/D';

[Walter Roberson]

For example,

format long g
C = randn(5,5)

C = 5×5

0.534075717580324 1.03784458298148 -0.616586361478136 0.295620475206361 0.334664464924375 -0.3252306031039 0.654350553093099 1.01272119034537 -2.91598774498595 0.490273123964035 -0.215032453113893 0.211591646656912 0.170684173870892 0.364278859645209 0.221115783934122 -0.168456938277056 0.232103674955222 -1.07419067480375 -0.934081265392528 -1.63007969245364 -0.594376546921606 0.226988230080239 -0.761402227083518 0.0139446077401544 1.40053255822073

D = randn(5,5)

D = 5×5

-1.19756792919788 -0.824243026868965 -2.50445524452914 0.101667066638342 -0.504703838460566 1.15150929259642 0.976772699910593 -1.25423360762766 -0.60771924982471 0.0321346566278941 -0.00510123137629798 -0.110274533571628 0.590246036756177 -0.59094462177991 0.237197946373215 1.61499789651319 1.16332794981143 -0.620320628103608 1.10345111916639 1.2998156700853 0.11723330386199 -0.287532426957904 -0.870333147694153 -0.347069635314334 0.00471197397717008

R = D\C/D'

R = 5×5

8.96323802608789 -9.32322181019618 0.410689824864551 2.50039750152997 -4.78328372751973 -11.0454559885476 10.9792555329219 -0.188115721627332 -4.01224895366599 5.65554582680323 -0.45689428204668 0.120491802180545 0.0343346029914989 0.0199323874611362 0.785785854837459 -0.156787750392925 0.336451574165717 0.0517879955590309 1.06971456156443 -0.560419450014348 -2.69331479250787 3.98666431049135 -0.16078984790504 0.161708121500883 0.277175380953083

and test

err = D * R * D' - C

err = 5×5

2.22044604925031e-15 -6.66133814775094e-16 -6.66133814775094e-16 5.55111512312578e-17 1.11022302462516e-15 -2.72004641033163e-15 4.44089209850063e-16 2.22044604925031e-16 1.77635683940025e-15 -2.05391259555654e-15 1.38777878078145e-16 -1.94289029309402e-16 0 -2.77555756156289e-16 5.55111512312578e-17 -2.74780198594726e-15 -4.44089209850063e-16 2.22044604925031e-16 0 -1.77635683940025e-15 -9.99200722162641e-16 -6.93889390390723e-16 -2.22044604925031e-16 6.64399091299117e-16 -2.22044604925031e-16

Those differences are obviously numeric round-off error that can be neglected.

[Ahmed Galal]

Thanks for you.

I tried your answer but all ellements of R were NaN

[Walter Roberson]

What is

size(C)
size(D)
rank(C)
rank(D)

[John D'Errico]

As importantly, what is the rank of D? Is D numerically singular? If it is, then unless C is ALSO singular, and has the same rank as D, AND C has some special properties, then no solution can exist.