Topologies on Quotient Space of Matrices via Semi-tensor Product

An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices leads to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.