## Subsection4.2.3Solving the normal equations

Let us review a method you have likely seen before for solving the LLS problem when matrix $A$ has linearly independent columns. We already used these results in Subsection 2.1.1

We wish to solve $A^H A \hat x = A^H b \text{,}$ where $A$ has linearly independent columns. If we form $B = A^H A$ and $y = A^H b \text{,}$ we can instead solve $B \hat x = y \text{.}$ Some observations:

• Since $A$ has linearly independent columns, $B$ is nonsingular. Hence, $\hat x$ is unique.

• $B$ is Hermitian since $B^H = (A^H A )^H = A^H (A^H)^H = A^H A = B \text{.}$

• $B$ is Hermitian Positive Definite (HPD): $x \neq 0$ implies that $x^H B x \gt 0 \text{.}$ This follows from the fact that

\begin{equation*} x^H B x = x^H A^H A x = ( A x )^H ( A x ) = \| A x \|_2^2. \end{equation*}

Since $A$ has linearly independent columns, $x \neq 0$ implies that $A x \neq 0$ and hence $\| A x \|_2^2 \gt 0 \text{.}$

In Section 5.4, you will find out that since $B$ is HPD, there exists a lower triangular matrix $L$ such that $B = L L^H \text{.}$ This is known as the Cholesky factorization of $B \text{.}$ The steps for solving the normal equations then become

• Compute $B = A^H A \text{.}$

Notice that since $B$ is Hermitian symmetric, only the lower or upper triangular part needs to be computed. This is known as a Hermitian rank-k update (where in this case $k = n$). The cost is, approximately, $m n^2$ flops. (See Subsection B.1.)

• Compute $y = A^H b \text{.}$

The cost of this matrix-vector multiplication is, approximately, $2 m n$ flops. (See Subsection B.1.)

• Compute the Cholesky factorization $B \rightarrow L L^H \text{.}$

Later we will see that this costs, approximately, $\frac{1}{3} n^3$ flops. (See Subsection 5.4.3.)

• Solve

\begin{equation*} L z = y \end{equation*}

(solve with a lower triangular matrix) followed by

\begin{equation*} L^H \hat x = z \end{equation*}

(solve with an upper triangular matrix).

Together, these triangular solves cost, approximately, $2 n^2$ flops. (See Subsection B.1.)

We will revisit this in Section 5.4.