Pseudoinverse and definiteness

The normal equations are a square $n\times n$ linear system to solve for $\mathbf{x}$ which leads to the defintion of the pseudoinverse as a formal solution:

$$\mathbf{A}^+ = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T$$

In practice this is not used for the same reason that the ordinary inverse is not used. But conceptually the \ operator is mathematically equivalent to left multiplying by the inverse (square matrix) or pseudoinverse (rectangular).

The matrix $\mathbf{A}^T\mathbf{A}$ has some important properties:

1. $\mathbf{A}^T\mathbf{A}$ is symmetric

2. $\mathbf{A}^T\mathbf{A}$ is singular only if the columns of $\mathbf{A}$ or linearly dependant.

3. If $\mathbf{A}^T\mathbf{A}$ is nonsingular, that it is positive definate.