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A Study on the Similarity of the Block Matrices

Cheng Shi-zhen

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Abstract

In this note, we show the two necessary and sufficient conditions such that two block matrices are similar, that is, suppose that the two square matrices A and B satisfy A2=0 and B2=0. Show that the two block matrices AC=0B and A0=0B are similar if and only if rank AC=0B=rank(A)+rank(B) and AC+CB=0. Suppose that the two square matrices A and B satisfy A2=A and B2=B. Show that the two block matrices AC=0B and A0=0B are similar if and only if AC+CB=C.

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What this paper is about

In this note, we show the two necessary and sufficient conditions such that two block matrices are similar, that is, suppose that the two square matrices A and B satisfy A2=0 and B2=0. Show that the two block matrices AC=0B and A0=0B are similar if and only if rank AC=0B=rank(A)+rank(B) and AC+CB=0. Suppose that the two square matrices A and B satisfy A2=A and B2=B. Show that the two block matrices AC=0B and A0=0B are similar if and only if AC+CB=C.

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Available abstract

In this note, we show the two necessary and sufficient conditions such that two block matrices are similar, that is, suppose that the two square matrices A and B satisfy A2=0 and B2=0. Show that the two block matrices AC=0B and A0=0B are similar if and only if rank AC=0B=rank(A)+rank(B) and AC+CB=0. Suppose that the two square matrices A and B satisfy A2=A and B2=B. Show that the two block matrices AC=0B and A0=0B are similar if and only if AC+CB=C.

Key concepts: Rank (graph theory), Block (permutation group theory), Combinatorics, Mathematics, Square (algebra), Similarity (geometry), Matrix (chemical analysis), Computer science

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