25.4: Existence and Uniqueness - General Case (Square Systems)

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Page ID: 55701

Masayuki Yano, James Douglass Penn, George Konidaris, & Anthony T Patera
Massachusetts Institute of Technology via MIT OpenCourseWare

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We now consider a general system of \(n\) equations in \(n\) unknowns, \[\underbrace{A}_{\text {given }} \underbrace{u}_{\text {to find }}=\underbrace{f}_{\text {given }}\] where \(A\) is \(n \times n, u\) is \(n \times 1\), and \(f\) is \(n \times 1\).

If \(A\) has \(n\) independent columns then \(A\) is non-singular, \(A^{-1}\) exists, and \(A u=f\) has a unique solution \(u\). There are in fact many ways to confirm that \(A\) is non-singular: \(A\) has \(n\) independent columns; \(A\) has \(n\) independent rows; \(A\) has nonzero determinant; \(A\) has no zero eigenvalues; \(A\) is SPD. (We will later encounter another condition related to Gaussian elimination.) Note all these conditions are necessary and sufficient except the last: \(A\) is \(\mathrm{SPD}\) is only a sufficient condition for non-singular \(A\). Conversely, if any of the necessary conditions is not true then \(A\) is singular and \(A u=f\) either will have many solutions or no solution, depending of \(f .-\) In short, all of our conclusions for \(n=2\) directly extend to the case of general \(n\).

\({ }^{6}\) Note in the computational context we must also understand and accommodate "nearly" singular systems. We do not discuss this more advanced topic further here.