In the decomposition of a symmetric matrix $A = \sum_{i=0}^{n} \lambda_i u_i u_i^\top$, what is the term $u_i u_i^\top$?

Question

I'm not sure what $u_i u_i^\top$ is. It looks like two vectors being multiplied, but that would produce a scalar (doesn't make sense that the sum of scalars is a matrix...). I don't know what to search for and so I can't even find the proof for this. Any help would be appreciated!

Answer 1

If vector $u$ is an $n$-dimensional column vector, $u u^T$ is an $n \times n$ matrix, with $(i,j)$ entry $u_i u_j$. Thus

$$ \pmatrix{u_1\cr u_2 \cr u_3} \pmatrix{u_1 & u_2 & u_3} = \pmatrix{u_1^2 & u_1 u_2 & u_1 u_3\cr u_2 u_1 & u_2^2 & u_2 u_3\cr u_3 u_1 & u_3 u_2 & u_3^2}$$