Let {u1, u2, . . . , un} be an orthonormal basis for Rn and let A be a linear combination of the rank 1 matrices u1uT1 , u2uT2 , . . . , unuTn . If A = c1u1uT1 + c2u2uT2 +· · ·+cnunuTn show that A is a symmetric matrix with eigenvalues c1, c2, . . . , cn and that ui is an eigenvector belonging to ci for each i.

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