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I'm trying to solve the following minimization problem: $$\underset{\lambda\in[a,b]}{\min}\|A(\lambda)X-X\|_2$$ for given $a<b$ in $\mathbb{R}$ and where $A(\lambda)$ is matrix of size $n\times n$ where entries depend on a parameter $\lambda$ and $X$ is a column vector of $\mathbb{R}^n$.

We have : $$\|A(\lambda)X-X\|_2=\|(A(\lambda)-I_n)X\|_2$$ And if $X$ is fixed during optimization and completely independant from $\lambda$, it turns out that solving : $$\underset{\lambda\in[a,b]}{\min}\|A(\lambda)X-X\|_2$$ is equivalent to solve : $$\underset{\lambda\in[a,b]}{\min}\|A(\lambda)-I_n\|_2$$ ?

With matrix norms, i can majorate with norm product of each terms involved and once done, the norm of $X$ is just a constant so i can remove it from the minimization process. What do you think about that , the same for "regular" norms?

Best !

ar_ma_cs
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Not in general. Consider $$A(\lambda)=\left[\begin{array}{cc}\lambda+1&-1\\0&1\end{array}\right].$$ If $X=(x,y)$ then $$A(\lambda)X-X=((\lambda x-y),0)$$ and the $\lambda$ minimizing the norm of that vector can clearly change with changing $X$ so it cannot be computed by any process completely independent of $X$.

Michal Adamaszek
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