operator symbol "max" in optimization

Question

The section 4.2 "Poor Conditioning" in the book Deep Learning defines the condition number of the function $f(x) = A^{-1}x$ as

\begin{align} \underset{i,j}{\max}~ \Bigg| \frac{\lambda_i}{ \lambda_j} \Bigg|. \end{align}

and explains

the ratio of the magnitude of the largest and smallest eigenvalue.

I understand the eigenvalue, the ratio and the magnitude part.

what does the operation symbol "max" refer to? Is it some kind of optimization operator?

score 2 · Accepted Answer · answered May 01 '20 at 05:52

2

"Max" here refers to the maximum of the magnitude of all the ratios of one eigen value to another.

answered May 01 '20 at 05:52

Thank you. What does the "$i,j$" under that operator mean? – JJJohn May 01 '20 at 05:55
"i,j" refers to $i^{th}$ eigenvalue and $j^{th}$ eigenvalue. Hope it helps you. – G R Krishna Chand Avatar May 01 '20 at 05:58
3

@shi95 $i,j$ tells you what you’re allowed to change to maximise what comes after $\max$. Suppose the eigenvalues are ${\lambda_1,\cdots,\lambda n}$, then it’s short for $\max{\lvert \lambda_i/\lambda_j\rvert : i,j\in{1,\dots,n}}$ – gen-ℤ ready to perish May 01 '20 at 06:56

1 Answers1