I am trying to read a paper which involves some linear algebra. I would appreciate if anyone could clarify what the below statements exactly mean.
1-) $0 \preceq B^TB \preceq A^TA$ where $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{l \times m}$ are matrices.
2-) $\langle A_i,x \rangle^2$ where $A_i$ and $x$ are vectors
On
I have seen the notation in part 1 to mean that every entry of $B^{T}B$ is greater than or equal to $0$ and in turn every entry in $A^{T}A$ is greater than or equal to the equivalent entry of $B^{T}B$. That is, $b_{ij} \leq a_{ij}$. This creates a partial ordering.
Note that both $A^{T}A$ and $B^{T}B$ are of dimension $m \times m$, so this is a well defined partial ordering.
So, I could say that $P ⪯ Q$ if $$P = \begin{bmatrix}1 & 2\\3 & 5\end{bmatrix}$$ and $$Q = \begin{bmatrix}4 & 3\\3 & 9\end{bmatrix}$$
I would assume that the angle brackets in part 2 just mean "inner product." Multiply the vectors together in the "usual way" (Assuming the Euclidean inner product) and then square (as per the exponent.)
A symmetric square matrix $M$ of size $m\times m$ is said to be positive $(0\preceq M)$ iff it is positive semidefinite, i.e. $x^TMx\ge 0$ for all vectors $x\in\Bbb R^m$.
And, we define $M\preceq M'$ iff $\ 0\,\preceq\, M'-M$.
I think, it is more suitable to this problem, as both $B^TB$ and $A^TA$ are $m\times m$ symmetric matrices.
If $x,y$ are vectors, specifically when $\in\Bbb R^m$, then $\langle x,y\rangle$ means their scalar (/inner/dot) product which coincides with the matrix product $x^Ty$.