Suppose that $1\le p \le \infty$, that $\vec{v}$ is a vector, that $k$ is a scalar, and that $A$ is a matrix.

Question

Suppose that $1\le p \le \infty$, that $\vec{v}$ is a vector, that $k$ is a scalar, and that $A$ is a matrix.

(a) Show that

$$\|k \vec{v}\|_p=|k|\|\vec{v}\|$$

(b) suppose $\lambda $ is an eigenvalue of $A.$ Show that

$$\|A\|_p\ge \lambda$$

for (a) lets consider $\vec{v}=(v_1,v_2,\cdots ,v_n)$ then

$\|kv\|_p=(\sum_i |kv_i|^p)^\frac{1}{p}=|k|(\sum_i |v_i|^p)^\frac{1}{p}=|k|\|v\|_p$

am I correct?

how to prove part(b).?