The matrix norm of the identity matrix disturbed by a small matrix

Question

So I am consider the norm of matrix $\|I-C\|_2$, where $C$ is a positive definite matrix with a very small norm, what can we say about $\|I-C\|_2$? Like, is it smaller than $1$? Or can I express it w.r.t $C$?

Thank you in advance!

Answer 1

Let $S_n$ be the set of $n\times n$ real symmetric matrices.

$X\in S_n\mapsto ||X||_2=\rho(X)$ is continuous but not differentiable in a neighborhood of some $X$ which has several eigenvalues $\lambda,\mu$ s.t. $|\lambda|=|\mu|=\rho(X)$.

Here $||I-C||_2=1-\lambda<1$ where $\lambda=\min(spectrum(C))$.

Answer 2

Since matrix $\rm C$ is symmetric and positive definite "with a very small norm",

$$\alpha \, \mathrm I \preceq \mathrm C \preceq \beta \, \mathrm I$$

where $0 < \alpha < \beta \ll 1$. Hence,

$$- \beta \, \mathrm I \preceq -\mathrm C \preceq - \alpha \, \mathrm I$$

and

$$(1 - \beta) \, \mathrm I \preceq \mathrm I - \mathrm C \preceq (1 - \alpha) \, \mathrm I$$

Thus,

$$\| \mathrm I - \mathrm C \|_2 = \sigma_{\max} \left( \mathrm I - \mathrm C \right) = \lambda_{\max} \left( \mathrm I - \mathrm C \right) = 1 - \lambda_{\min} \left( \mathrm C \right) = 1 - \alpha < 1$$

which agrees with Loup Blanc's answer.