In this paper, we prove several results which are generalizations of basic facts of spectral radius. Among other inequalities, we prove that if \(A\in \mathbb {M}_m(\mathbb {M}_n)\) , where \(\mathbb {M}_m(\mathbb {M}_n)\) is the set of all \(m*m\) block complex matrices with each block in \(\mathbb {M}_n(\mathbb { C})\) . Then \(\begin{aligned} r^{(2)}(A) = \displaystyle \lim _{h \rightarrow \infty } (||A^{\circ ^h}||^{(2)})^{{ \circ }^{\frac{1}{h}}} \end{aligned}\) and \(\begin{aligned} r^{(1)}(A) =\displaystyle \lim _{h \rightarrow \infty } (||\widetilde{ \tilde{A} ^{\circ ^h}}||^{(1)})^{{\circ }^{\frac{1}{h}}}, \end{aligned}\) where \(r^{(1)}(A)\) and \(||A||^{(1)}\) are, respectively, the first partial matrix of spectral radius and the first partial matrix of spectral norm which are defined in this paper. \(r^{(2)}(A)\) and \(||A||^{(2)}\) are, respectively, the second partial matrix of spectral radius and the second partial matrix of spectral norm.
