I've been told that I can write a square matrix $A$ like this: $\lambda_1 V_1 V_1^{T} + \lambda_2 V_2 V_2^{T} + ... \lambda_n V_n V_n^{T}$, i.e., as an linear combinations of eigenvalues and the outer product of their associated eigenvectors ($V_i$ is unitarian), but for example, the following matrix:
[ 1. 0.5 -0.1]
[ 0.5 1. 10. ]
[ 2. 3. 5. ]
Have these eigenvalues: 8.93009292, 0.69188007 and -2.62197299 and the respective eigenvectors [ 0.04161113 0.78386684 0.61953313], [-0.0262233 0.94920296 0.3135699 ] and [ 0.03123074 0.91609679 0.39973905].
But doing the linear combination as above I get this:
[ 0.01433413 0.37633714 0.26041067]
[ 0.37633714 3.70535792 3.80969252]
[ 0.26041067 3.80969252 3.28030794]
This is nothing near the original matrix, what am I doing wrong?