Matrix expansion does not decrease norms

Question

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean norm of any sub-block $A_{ij}$, for $i,j = \left\{ {1,2} \right\}$ satisfy the inequality $${\left\| {{A_{ij}}} \right\|_2} \le {\left\| A \right\|_2}$$ and therefore the matrix expansion does not decrease norms? where can i find some reference of this proof if it is true.

Answer 1

It seems that $(∥A∥_2 )^2 = \sum_{k,l = 1}^N ∥a_{kl}∥^2 \geq \sum_{k,l \in S}∥a_{kl}∥^2 =∥(A_{ij}∥_2)^2$ simply because the subset $S \subsetneq \{1, 2, ...,N \}\times \{1, 2, ...,N \} $ ($S$ labels less or at most the same positive numbers) The matrix $A_{ij}$ contains the elements $a_{kl}$ for all $(k,l) \in S$. So what you said is always true.