If I have a matrix A = $\alpha$I where I is the identity matrix and a is some non-zero real number.
How would I go about taking the induced matrix norm of it?
I'm leaning towards using the submultiplicativity property, but I'm just not really sure how to approach this one, new to norms ;p. thanks
I don't believe that the submultiplicativity is needed here. Indeed, an induced matrix norm is by definition, shaped as $$\||M\|| = \sup_{x\neq 0}\frac{\|Ax\|}{\|x\|}$$ where $M$ is squared matrix, i.e. mapping vectors from a vector space $V$ to vectors of $V$, and $\|\cdot\|$ is a norm on $V$. Now, by homogeneity of the norm $\|\cdot \|$ we have that $\|\alpha I x\|=|\alpha | \|Ix\|=|\alpha | \|x\|$, and therefore, if $A=\alpha I$, for every $x\neq 0, x \in V$, we have $$\frac{\|Ax\|}{\|x\|}=\frac{|\alpha| \|x\|}{\|x\|}=|\alpha|,$$ implying $\||A\||=|\alpha|$.