# Find eigenvalues of specific matrix

Let the entries of $$n \times n$$ matrix $$A$$, where $$n \ge 3$$, be $$a_{ij} = \frac{((-1)^{i+j}+1)}{2}$$ Find all eigenvalues and multiplicity of $$A$$.

By nullity I know that $$\lambda=0$$ has multiplicity $$n-2$$, but the two other eigenvalues are hard to calculate. I know that those two eigenvalues sum up with $$n$$ by the trace. How could I calculate the rest eigenvalues without just predicting the eigenvalue? (E.g. Plugging $$\lambda=\frac{n}{2}$$)

• "By nullity, I know $\lambda=0$ has multiplicity $2$" How? What is the rank of $A$? Jun 2 '20 at 3:11
• @above that's a typo. It's $n-2$ Jun 2 '20 at 3:50
• Ok thanks for the clarificaiton Jun 2 '20 at 4:12

Let $$v=\left(-1,1,-1,1,\ldots,(-1)^n)\right)^T$$ and $$e=(1,1,\ldots,1)^T$$. Then $$A=\frac{vv^T+ee^T}{2}=\frac12\pmatrix{v&e}\pmatrix{v&e}^T$$. Therefore $$A$$ has the same multi-set of nonzero eigenvalues as $$B=\frac12\pmatrix{v&e}^T\pmatrix{v&e}=\frac12\pmatrix{n&\frac{(-1)^n-1}{2}\\ \frac{(-1)^n-1}{2}&n},$$ i.e. its eigenvalues are $$\frac12\left[n\pm\frac{(-1)^n-1}{2}\right]$$ and $$n-2$$ copies of zeros.