# sum of a non singular symmetric matrix and the matrix of its eigenvalues is invertible

I am looking for a proof for this lemma: Assume $\Phi$ is a real, symmetric and non-singular matrix of order $T\times T$ with non-negative elements; Let define $B=(u_1,...,u_T)$ as a column matrix of the eigenvectors of $\Phi$ and $\Gamma_T=diag(\gamma_1,...,\gamma_T)$ is a diagonal matrix that consists of the corresponding eigenvalues of $\Phi$. Then $\Phi+\Gamma_T$ is an invertible matrix.

This is wrong, with or without the hypothesis of non-negative entries (which was added to the question after I originally posted this answer). Take $$Φ=(√2/211√2/2)\Phi=\pmatrix{\sqrt2/2&1\\1&\sqrt2/2}$$ and $$B=(111−1)B=\pmatrix{1&1\\1&-1}$$, then $$ΓT=(√2/2+100√2/2−1)\Gamma_T=\pmatrix{\sqrt2/2+1&0\\0&\sqrt2/2-1}$$ and $$Φ+ΓT=(√2+111√2−1)\Phi+\Gamma_T=\pmatrix{\sqrt2+1&1\\1&\sqrt2-1}$$ is non invertible.