A new proof of the Perron-Frobeniuos theorem, a variational approach

We generalized the Perron-Frobenious theory by using a variational approach and extended it to sets of arbitrary matrices, including those that are not either irreducible or essentially positive. We introduce a new concept called a "quasi-eigenvalues of a matrix", which is invariant under orthogonal transformations of variables, and has various useful properties, including such as determining the largest of the real parts of eigenvalues of a matrix. Additionally, we establish new results on the continuity of the spectral radius, as well as obtain new types of estimates for the ranges of the sets of eigenvalues and their real parts.