Matrices of the form $A=(x_i x_j)_{i,j}$

Question

For some matrices that have a special form (e.g. Vandermonde matrices) there are simple, explicit expressions for their inverse available. A situation I ran into recently deals with matrices of the type $$A=(x_ix_j)_{i,j}=x\cdot x^T $$ where $x$ is a vector in $\mathbb{R}^n$. Are there similar formulas available?

Answer 1

The rank of $x\cdot x^t$ is at most $1$ because every column is a multiple of $x$, so it's not invertible.