Outer product

Let $$u$$ be a column vector. Then, $$u u^T$$ is called outer product. It is also called dyad or (symmetric) rank-one matrix.

If the columns of a matrix $$U = u_1, \dots, u_k$$ form a set of orthogonal vectors, then the $$U U^T$$ is a projection matrix. $$U U^T$$ can be written as (see Matrix multiplication)

$$U U^T = \sum_{i=1}^n u_i u_i^T$$

In this interpretation, each component $$q_i q_i^T$$ is an outer product and also a 1-dimensional projection matrix. This interpretation is illustrated with an example in Symmetric matrices.