2021-07-14

NOYAFA

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**An overview of q&a tester**

**Block Cholesky decomposition of q&a tester**

where the matrix A displaystyle beginmatrixAendmatrix is assumed to be non-singular, I displaystyle beginmatrixIendmatrix is an identity matrix with proper dimension, and 0 displaystyle beginmatrix0endmatrix is a matrix whose elements are all zero.

We can also rewrite the above equation using the half matrices: ( A B C D ) ( A 1 2 C A 1 2 ) ( A 1 2 A 1 2 B ) ( 0 0 0 Q 1 2 ) ( 0 0 0 Q 1 2 ) , displaystyle beginpmatrixA&BC&DendpmatrixbeginpmatrixA^frac 12CA^-frac 12endpmatrixbeginpmatrixA^frac 12&A^-frac 12Bendpmatrixbeginpmatrix0&00&Q^frac 12endpmatrixbeginpmatrix0&00&Q^frac 12endpmatrix,

where the Schur complement of A displaystyle beginmatrixAendmatrix

in the block matrix is defined by Q D C A 1 B displaystyle beginmatrixQD-CA^-1Bendmatrix

and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition.

The half matrices satisfy that A 1 2 A 1 2 A ; A 1 2 A 1 2 I ; A 1 2 A 1 2 I ; Q 1 2 Q 1 2 Q . displaystyle beginmatrixA^frac 12,A^frac 12A;endmatrixqquad beginmatrixA^frac 12,A^-frac 12I;endmatrixqquad beginmatrixA^-frac 12,A^frac 12I;endmatrixqquad beginmatrixQ^frac 12,Q^frac 12Q.endmatrix

Thus, we have ( A B C D ) L U , displaystyle beginpmatrixA&BC&DendpmatrixLU,

where L U ( A 1 2 0 C A 1 2 0 ) ( A 1 2 A 1 2 B 0 0 ) ( 0 0 0 Q 1 2 ) ( 0 0 0 Q 1 2 ) . displaystyle LUbeginpmatrixA^frac 12&0CA^-frac 12&0endpmatrixbeginpmatrixA^frac 12&A^-frac 12B0&0endpmatrixbeginpmatrix0&00&Q^frac 12endpmatrixbeginpmatrix0&00&Q^frac 12endpmatrix.

The matrix L U displaystyle beginmatrixLUendmatrix can be decomposed in an algebraic manner into L ( A 1 2 0 C A 1 2 Q 1 2 ) a n d U ( A 1 2 A 1 2 B 0 Q 1 2 ) . displaystyle LbeginpmatrixA^frac 12&0CA^-frac 12&Q^frac 12endpmatrixmathrm and UbeginpmatrixA^frac 12&A^-frac 12B0&Q^frac 12endpmatrix.

**Block LDU decomposition of q&a tester**

This may be useful for inversion if also D C A 1 B textstyle D-CA^-1B is non-singular: [ A B C D ] 1 [ I A 1 B 0 I ] 1 [ A 0 0 D C A 1 B ] 1 [ I 0 C A 1 I ] 1 [ I A 1 B 0 I ] [ A 0 0 D C A 1 B ] 1 [ I 0 C A 1 I ] displaystyle beginbmatrixA&BC&Dendbmatrix^-1beginbmatrixI&A^-1B0&Iendbmatrix^-1beginbmatrixA&00&D-CA^-1Bendbmatrix^-1beginbmatrixI&0CA^-1&Iendbmatrix^-1beginbmatrixI&-A^-1B0&IendbmatrixbeginbmatrixA&00&D-CA^-1Bendbmatrix^-1beginbmatrixI&0-CA^-1&Iendbmatrix

An equivalent UDL decomposition exists if D textstyle D is non-singular: [ A B C D ] [ I B D 1 0 I ] [ A B D 1 C 0 0 D ] [ I 0 D 1 C I ] displaystyle beginbmatrixA&BC&DendbmatrixbeginbmatrixI&BD^-10&IendbmatrixbeginbmatrixA-BD^-1C&00&DendbmatrixbeginbmatrixI&0D^-1C&Iendbmatrix

This may be useful for inversion if A B D 1 C textstyle A-BD^-1C is non-singular: [ A B C D ] 1 [ I 0 D 1 C I ] 1 [ A B D 1 C 0 0 D ] 1 [ I B D 1 0 I ] 1 [ I 0 D 1 C I ] [ A B D 1 C 0 0 D ] 1 [ I B D 1 0 I ] displaystyle beginbmatrixA&BC&Dendbmatrix^-1beginbmatrixI&0D^-1C&Iendbmatrix^-1beginbmatrixA-BD^-1C&00&Dendbmatrix^-1beginbmatrixI&BD^-10&Iendbmatrix^-1beginbmatrixI&0-D^-1C&IendbmatrixbeginbmatrixA-BD^-1C&00&Dendbmatrix^-1beginbmatrixI&-BD^-10&Iendbmatrix

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