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 Φ is a real, symmetric and non-singular matrix of order T×T with non-negative elements; Let define B=(u1,...,uT) as a column matrix of the eigenvectors of Φ and ΓT=diag(γ1,...,γT) is a diagonal matrix that consists of the corresponding eigenvalues of Φ. Then Φ+ΓT is an invertible matrix.

Answer

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=(2/2112/2) and B=(1111), then ΓT=(2/2+1002/21) and Φ+ΓT=(2+11121) is non invertible.

Attribution
Source : Link , Question Author : karo solat , Answer Author : Marc van Leeuwen

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