I am civil engineering student , and I don't have any information about positive definite, rank, and conjugate . Please help me if you can.
I have an algorithm that finds the minimum.
I have a matrix $H$, and $H$ be an $n\times n$ symmetric matrix, and $$ f(x)= c^Tx+\frac12x^THx.$$ Suppose $D_1$ is a $n×n$ positive definite symmetric matrix, let $x_1$ be an initial point. For $j=1,..,n$ let $\lambda_j= \arg\min_{λ\ge0} f(x+ \lambda d_j)$, and $x_{j+1}=x_j+ \lambda_jd_j$ where $d_j=-D_j\nabla f(x_j) $ and $D_{j+1}= D_j+ a_j$
$$a_j=\frac{(p_j-D_jq_j)(p_j-D_jq_j)^T}{q_j^T(p_j-D_jq_j)}$$
$p_j=x_{j+1}-x_j$ and $q_j=Hp_j$.
In this algorithm, why is the matrix $a_j$ of rank $1$?
If $D_j$ be a positive definite, is necessarily $D_{j+1}$ positive definite too?
Are the direction $d_1,...,d_n$ necessarily conjugate?
Thanks
The matrix $a_j$ is rank $1$ since it is a matrix of the form $uv^T$, where $u$ and $v$ are column vectors. Note that in such a matrix, every row will be a multiple of $v^T$.
The matrix $a_j$ is symmetric, and it will also be positive semidefinite since if the number on bottom is positive. In this case, we can say that $a_j$ is positive semidefinite since it can be written in the form $a_j = MM^T$. In particular, we have $$ a_j = \left[\frac{p_j - D_j q}{\sqrt{q_j^T(p_j - D_jq_j)}}\right]\left[\frac{p_j - D_j q}{\sqrt{q_j^T(p_j - D_jq_j)}}\right]^T $$
If both $D_j$ and $a_j$ are positive semidefinite, then their sum $a_j + D_j$ will also be positive semidefinite.