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

Question 1

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.

Question 2

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 $\Phi=\pmatrix{\sqrt2/2&1\\1&\sqrt2/2}$ and $B=\pmatrix{1&1\\1&-1}$ , then $\Gamma_T=\pmatrix{\sqrt2/2+1&0\\0&\sqrt2/2-1}$ and $\Phi+\Gamma_T=\pmatrix{\sqrt2+1&1\\1&\sqrt2-1}$ is non invertible.

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

Answer

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