Eigenvalues of a tridiagonal matrix with $-1, 2, -1$ as entries

Asked Oct 21 '20 at 13:29

Active Oct 21 '20 at 13:29

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This problem was given as an exercise for section 6.2 of Hoffman & Kunze's book in a course in linear algebra, which I couldn't solve and I really don't know how to solve it, may be because I don't know where does that $\sin^2$ come from.

Also I searched for it on the internet and results weren't close to this question and I tought may be it's wrong.

Anyway, this is the problem:

If $$ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 & -1 \\ & -1 & 2 & -1 \\ && -1 & 2 & -1 \\ &&& -1 & 2 & -1 \\ &&&& -1 & 2 \end{bmatrix} $$ and $A \in M_n(\mathbb{C})$, then show eigenvalues of A are: $$4\sin^2 \left( \frac {j \pi}{2n+2} \right)$$ for $1\le j \le n$.

Any help is appriciated.

asked Oct 21 '20 at 13:29

Masoud

3

See Davide Giraudo's answer here: https://math.stackexchange.com/questions/177957/eigenvalues-of-tridiagonal-symmetric-matrix-with-diagonal-entries-2-and-subdiago?rq=1 – v_lentin Oct 21 '20 at 13:40
@v_lentin Thanks. But I'm not really happy with using recurrence relations. Do you know any other strategy to solve the problem? – Masoud Oct 21 '20 at 18:02
@Masoud You might prefer the answer given here then – Ben Grossmann Oct 21 '20 at 19:00

Eigenvalues of a tridiagonal matrix with $-1, 2, -1$ as entries

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