12. Eliminate one block and recover the solution
All examples · Glossary: Schur complement · Papers, proofs, Lean, and code
Solve the real system
\[\begin{pmatrix}2&1\\3&4\end{pmatrix}
\begin{pmatrix}u\\v\end{pmatrix}
=\begin{pmatrix}5\\11\end{pmatrix}.
\]
Here each block is a scalar: \(A=2\), \(B=1\), \(C=3\) and \(D=4\).
Eliminate the first unknown
The first equation gives \(u=(5-v)/2\). Substitute this into the second:
\[3(5-v)/2+4v=11,\qquad
(4-3/2)v=11-15/2=7/2.
\]
The reduced coefficient is the Schur complement
\[S=D-CA^{-1}B=4-3\cdot(1/2)\cdot1=5/2.
\]
Since \(S\) is nonzero, \(v=(7/2)/(5/2)=7/5\). Back substitution gives \(u=(5-7/5)/2=9/5\).
Check both original equations
\[2(9/5)+7/5=25/5=5,\qquad
3(9/5)+4(7/5)=55/5=11.
\]
The matrix factorization displays the same elimination:
\[\begin{pmatrix}2&1\\3&4\end{pmatrix}
=\begin{pmatrix}1&0\\3/2&1\end{pmatrix}
\begin{pmatrix}2&1\\0&5/2\end{pmatrix}.
\]
Its determinant is \(2\cdot4-1\cdot3=5\), also the product \(2(5/2)\).
Keep the invertibility assumptions
For compatible square diagonal blocks, the same substitution gives
\[Sv=g-CA^{-1}f,\qquad u=A^{-1}(f-Bv).
\]
This form requires invertible \(A\). A unique solution for every right-hand side additionally requires invertible \(S\). If \(S\) is singular, a particular right-hand side can give no solution or multiple solutions.
For example, replace \(D=4\) by \(D=3/2\). The reduced coefficient becomes zero. With \(f=5\), consistency requires \(g=15/2\). If \(A\) itself is singular, this elimination route is unavailable, although a different pivot may work.
Try it
Keep the original matrix and use the right-hand side \((3,5)^{\mathsf T}\).
Answer: \((5/2)v=5-9/2=1/2\), so \(v=1/5\) and \(u=7/5\). Substitution gives \(2(7/5)+1/5=3\) and \(3(7/5)+4/5=5\).
Papers, proofs, Lean, and code
Terms in this example: Block matrix · Schur complement · Elimination · Determinant · Inverse
Paper VI gives elimination background. The collected Gaussian elimination guide retains its pending source qualification and unassessed confidence.
Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations
Read · Written proofs · Lean coverage · Code and setup · Evidence
Gaussian Elimination: collected guide
Read · Written proofs · Lean coverage · Code and setup · Evidence
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: block_elimination(). This checks the displayed rational, modular or finite-set calculations exactly.
The Lean links open each work’s recorded coverage and sources. They do not certify this tutorial. A work may have limited coverage or no linked Lean or code. Its page states what is available.