Hainsworth theorem

The Hainsworth theorem is a statement about the properties of Schur additions of three consecutively nested matrices.

Formulation

Let be $A={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}$ ${\ displaystyle A = {\ begin {pmatrix} A_ {11} & A_ {12} & A_ {13} \\ A_ {21} & A_ {22} & A_ {23} \\ A_ {31} & A_ {32} & A_ {33 } \ end {pmatrix}}}$ $A={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}$ , $B={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}$ ${\ displaystyle B = {\ begin {pmatrix} A_ {11} & A_ {12} \\ A_ {21} & A_ {22} \ end {pmatrix}}}$ $B={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}$ , $C=A_{11}$ ${\ displaystyle C = A_ {11}}$ $C=A_{11}$ - square matrix, and matrix $B$ ${\ displaystyle B}$ $B$ and $C$ ${\ displaystyle C}$ $C$ are nondegenerate. Matrix Shura Supplement $C$ ${\ displaystyle C}$ $C$ in the matrix $B$ ${\ displaystyle B}$ $B$ $\left(B{\mathcal {j}}C\right)=A_{22}-A_{21}A_{11}^{-1}A_{12}$ ${\ displaystyle \ left (B {\ mathcal {j}} C \ right) = A_ {22} -A_ {21} A_ {11} ^ {- 1} A_ {12}}$ $\left(B{\mathcal {j}}C\right)=A_{22}-A_{21}A_{11}^{-1}A_{12}$ can be considered as a matrix submatrix $\left(A{\mathcal {j}}C\right)={\begin{pmatrix}A_{22}&A_{23}\\A_{32}&A_{33}\end{pmatrix}}-{A_{21} \choose A_{31}}A_{11}^{-1}(A_{12}A_{13})$ ${\ displaystyle \ left (A {\ mathcal {j}} C \ right) = {\ begin {pmatrix} A_ {22} & A_ {23} \\ A_ {32} & A_ {33} \ end {pmatrix}} - {A_ {21} \ choose A_ {31}} A_ {11} ^ {- 1} (A_ {12} A_ {13})}$ $\left(A{\mathcal {j}}C\right)={\begin{pmatrix}A_{22}&A_{23}\\A_{32}&A_{33}\end{pmatrix}}-{A_{21} \choose A_{31}}A_{11}^{-1}(A_{12}A_{13})$ .

Then:

\left(A{\mathcal {j}}B\right)=\left(\left(A{\mathcal {j}}C\right){\mathcal {j}}\left(B{\mathcal {j}}C\right)\right)

{\ displaystyle \ left (A {\ mathcal {j}} B \ right) = \ left (\ left (A {\ mathcal {j}} C \ right) {\ mathcal {j}} \ left (B {\ mathcal {j}} C \ right) \ right)}

\left(A{\mathcal {j}}B\right)=\left(\left(A{\mathcal {j}}C\right){\mathcal {j}}\left(B{\mathcal {j}}C\right)\right)

.

Proof

The proof is in the book ^[1] .

Notes

↑ Problems and theorems of linear algebra, 1996 , p. 32.

Literature

Prasolov V. V. Problems and theorems of linear algebra. - M .: Science, 1996. - 304 p.

[_f2b600756d341805-1] Problems and theorems of linear algebra, 1996 , p. 32.

Hainsworth theorem

Formulation

Proof

Notes

Literature

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