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For $T>0$ and the constant matrices over the real numbers in the Kalman rank criterion, the finite-time Controllability Gramian accumulates the influence of all command times. Its column space equals that of the controllability matrix, and $W_T$ is a symmetric positive-definite matrix exactly when the controllability matrix has rank of a matrix $n$. If so, put $\delta =x_1-e^{AT}{x}_0$. Among square-integrable functions steering $x_0$ to $x_1$ at $T$, the commandminimizes ${\int }_0^T\Vert u(t){\Vert }^{2}dt$, with value ${\delta }^T{W}_T^{-1}\delta$. Here $T$ in a matrix superscript denotes transpose. This cost is a squared input norm, not necessarily physical energy, and the optimizing command need not satisfy an amplitude bound. ^[ref]

For the double integrator, $e^{As}B=(s,1{)}^T$, so direct integration givesFor a rest-to-rest distance $d$, $u_{\ast }(t)=6d/T^2-12dt/T^3$ and the minimum squared input norm is $12d^2/T^3$. Compare the two constant commands in the double integrator example, whose cost is $16d^2/T^3$: both attain the target, but the smooth command costs less. At $T=2$, $d=1$, however, $u_{\ast }$ peaks at $3/2$, so it violates $|u|\le 1$ while the two constant commands remain admissible. Controllability, optimal control, and constrained reachability in control theory answer different questions.

Consider the original small Poisson equation exampleIts exact solution is the quadratic polynomial $u(x)=1+2x-x^2$. Use a uniform mesh $x_i=ih$ with $h=1/4$ and three interior unknowns $U_1,U_2,U_3$. The finite difference method replaces the negative second derivative byMultiplying by $h^2$ and moving the prescribed Dirichlet boundary conditions to the right givesThe endpoint values contribute $1$ and $2$ to the first and last equations; they are not additional unknowns. The Thomas algorithm givesand back substitution yieldsAll three values equal $u(x_i)$ exactly: the centered finite difference for the second derivative is exact on a quadratic polynomial. This special nodal exactness does not assert exactness for arbitrary forcing or for the piecewise-linear curve joining the nodes.

For any $n$ interior points with $h=1/(n+1)$, the scaled tridiagonal matrix has $b_i=2$ and $a_i=c_i=-1$. Its pivot elements are $p_i=(i+1)/i$: start at $p_1=2$ and use $p_i=2-1/p_{i-1}$. All are positive. Interior rows have equality in diagonal dominance, so strict diagonal dominance is not the reason this example works. Instead it is a symmetric positive-definite matrix, since for a real vector $v\ne 0$ with $v_0=v_{n+1}=0$,Zero would force every neighbouring value equal and hence all values zero. This connects the boundary constraints directly to the safe pivots for tridiagonal elimination.

The displayed Thomas algorithm requires every modified pivot element to be nonzero in exact arithmetic. Nonzero original diagonal entries alone are insufficient: later pivot elements can vanish after elimination. Two useful sufficient contexts are a real symmetric positive-definite matrix, and strict diagonal dominance by rows:For the symmetric positive-definite matrix case, elimination leaves positive-definite remaining blocks by the Schur complement, so each $p_i>0$. Equivalently, the initial principal minors ${\Delta }_i=\det A_{1:i,1:i}$ are positive, and $p_i={\Delta }_i/{\Delta }_{i-1}$ with ${\Delta }_0=1$.

For strict diagonal dominance, a short induction shows $|p_i|>|c_i|$. The first row gives $|p_1|=|b_1|>|c_1|$. Assuming the preceding bound, the triangle inequality givesIn particular $p_n\ne 0$. These are structural guarantees against a zero pivot element, not universal bounds on forward error in floating-point arithmetic. A poorly conditioned linear system can still amplify roundoff error; numerical stability and the condition number must be distinguished.