Signature of a quadratic form

Symmetric matrix.
$$A = \frac12\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.$$

Eigenvalues. The eigenvalues of the matrix $\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}$ are $2$ (eigenvector $(1,1,1)$) and $-1$ (multiplicity $2$, eigenspace $x+y+z=0$). So the eigenvalues of $A$ are $1$ (multiplicity $1$) and $-1/2$ (multiplicity $2$).

Signature. One positive, two negative. Signature: $(1,2,0)$.

Remark. In the basis $(1,1,1)$, $(1,-1,0)$, $(1,0,-1)$, the form becomes $u^2 - \frac12 v^2 - \frac12 w^2$.