3Solving the Matrix Equation
3.1 Matrix Inversion
The last chapter discussed how the interpret the matrix equation
and established that the least squares normal equations have this form. The next question to address is, knowing
and
, how to discover
? And indeed, to determine whether an
exists to satisfy (3.1) and, if so, whether it is unique? All these questions can now be answered.
The first requirement is to define a special class of matrices. The identity matrix of dimension
is the square matrix having 1s in the diagonal positions and zeros elsewhere. In generic form, this is written

The subscript indicating the dimension is often omitted when the context is clear.
The special feature of the identity matrix is that when it pre‐ or postmultiplies a conformable matrix or vector, it leaves ...