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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?

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  • $\begingroup$ example $A=\begin{bmatrix}\frac{1}{2} & 0 \\ 0& \frac{1}{2}\end{bmatrix}$ $\endgroup$
    – JJR
    May 16 '17 at 12:52
  • $\begingroup$ For a process take any $B\neq 0$ and define $A:= \frac{B}{||B||_2+\varepsilon}$, $\varepsilon>0$. $\endgroup$
    – JJR
    May 16 '17 at 12:54
  • $\begingroup$ Thanks! Why does this work? Any reference? $\endgroup$
    – N8_Coder
    May 16 '17 at 13:00
  • $\begingroup$ What do you mean by why? $\endgroup$
    – JJR
    May 16 '17 at 13:02
  • $\begingroup$ Why is it guaranteed that this process gives an A with $\vert \vert A \vert \vert_2 <1 $? $\endgroup$
    – N8_Coder
    May 16 '17 at 13:18

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