= Sine eigenvectors of the fixed-end spring chain
For $N$ identical masses between fixed walls with equal springs, the dimensionless stiffness matrix has diagonal entries $2$ and adjacent entries $-1$. Its normalized <eigenvectors> and <eigenvalues> are
$$
a_i^{(j)}=\sqrt{\frac{2}{N+1}}\sin\frac{ij\pi}{N+1},\qquad
\lambda_j=2-2\cos\frac{j\pi}{N+1},\qquad 1\leq i,j\leq N.
$$
Extend each vector by $a_0=a_{N+1}=0$. The sine addition identity gives $2a_i-a_{i-1}-a_{i+1}=\lambda_j a_i$, including the fixed-end rows. The finite sine <orthogonality> identity supplies the normalization and makes distinct vectors orthogonal. The <normal mode> angular frequencies are $\omega_j=\sqrt{k\lambda_j/m}$ for mass $m$ and spring constant $k$. This complements the <fixed-end spring-chain determinant and its repeated-root limit> with the spatial mode shapes.
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