inverseSuperMatrix -- The inverse of a super matrix.

Usage:

N = inverseSuperMatrix(G, R)
Inputs:
- G, an instance of the type SuperMatrix,
- R, a ring,
Outputs:
- M, a matrix,

Description

A super Matrix $M={{M1, M2}, {M3, M4}}$ is invertible, if both the diagonal blocks, $M_1$ and $M_4$ are invertible.

In this case, the inverse is given by a blocked matrix, $T=\begin{pmatrix} T1&T2\\ T3&T4\end{pmatrix}$, where $T_1=(M_1 − M_2M^{-1}_4 M_3)^{-1}$, $T_2=−M^{-1}_1 M_2(M_4 − M_3M^{-1}_1 M_2)^{-1}$, $T_3=−M^{-1}_4 M_3(M_1 − M_2M^{-1}_4 M_3)^{-1}$, and $T_4=(M_4 − M_3M^{-1}_1 M_2)^{-1}$.

i1 : M1 = matrix{{5, 7}, {1, 2}};

              2       2
o1 : Matrix ZZ  <-- ZZ

i2 : M2 = matrix{{1, 2, 3}, {4, 5, 6}};

              2       3
o2 : Matrix ZZ  <-- ZZ

i3 : M3 = matrix{{3, 4}, {5, 6}, {7, 8}};

              3       2
o3 : Matrix ZZ  <-- ZZ

i4 : M4 = matrix{{2, 3, 11}, {4, 5, 6}, {7, 8, 9}};

              3       3
o4 : Matrix ZZ  <-- ZZ

i5 : G = superMatrixGenerator(M1, M2, M3, M4);

i6 : inverseSuperMatrix(G, QQ)

o6 = | -1 -3/4  0    11/4    -1     |
     | 1  1/2   0    -5/2    1      |
     | 1  5/84  1/7  -139/28 58/21  |
     | -1 11/84 -2/7 131/28  -53/21 |
     | 0  -1/42 1/7  -3/14   2/21   |

              5       5
o6 : Matrix QQ  <-- QQ

Caveat

Ways to use inverseSuperMatrix:

inverseSuperMatrix(SuperMatrix,Ring)

For the programmer

The object inverseSuperMatrix is a method function.

The source of this document is in SuperLinearAlgebra.m2:645:0.