# How to construct a matrix with given l2 norm requirement?

Can somebody tell me how to construct a matrix satisfying $\vert \vert A \vert \vert_2 <1$?

Note that the norm is defined as $\vert \vert A \vert \vert_2 = \sqrt{ \lambda_{max} (A^T A)}$.

Are there any good theorems making this design process deterministic?

• example $A=\begin{bmatrix}\frac{1}{2} & 0 \\ 0& \frac{1}{2}\end{bmatrix}$
– JJR
May 16 '17 at 12:52
• For a process take any $B\neq 0$ and define $A:= \frac{B}{||B||_2+\varepsilon}$, $\varepsilon>0$.
– JJR
May 16 '17 at 12:54
• Thanks! Why does this work? Any reference? May 16 '17 at 13:00
• What do you mean by why?
– JJR
May 16 '17 at 13:02
• Why is it guaranteed that this process gives an A with $\vert \vert A \vert \vert_2 <1$? May 16 '17 at 13:18