Solve $Lx = b$ for $x$ when $b = (1, 1, 2)^T$.

Let $L$ be a $3×3$ symmetric matrix with eigenvalues $λ_1 = −1$, $λ_2 = −2$, and $λ_3 = −3$ and their corresponding egenvectors $v_1 = (1, 1, 1)^T$, $v_2 = (2, −1, −1)^T$, and $v_3 = (0, 1, −1)^T$ where $T$ denotes the transpose.

Solve $Lx = b$ for $x$ when $b = (1, 1, 2)^T$.

Note: this is the idea behind Sturm-Liouville series solutions of ODE boundary value problems.
Ordinary Differential Equations Partial Differential Equations Differential Equations
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Martin Martin
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