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Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
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Bandwidth reduction of multiple matrices
Suppose I have a symmetric matrix $A$ and several diagonal matrices $D_1,D_2,\dots$. Are there any matrix transformations such as $P^\top A P$ so that
$$P^\top A P, P^\top D_1 P, P^\top D_2 P, \dot …