A Matrix Identity And Its Applications

The rational canonical form of matrix plays a very important role in linear alge-bra. An arbitrary matrix A is similar to a Jordan form matrix on complex field, but an arbitrary matrix is similar to rational canonical on general field. Domestic scholars Heguo Liu and Jilin Qian have proved an inequality about multiplicity of the matrix characteristic via rational canonical form theory [2]. One year later, Heguo Liu and domestic scholar Yun Fan have generalized the above inequality [3]. In this paper, we have proved a matrix identity and given several applications of it as following:1) We will prove the identity of matrix over the field F by induction on r: C and N are r×r matrices,I is a r×r identity matrix.2) Using the above identity and rational canonical form theory to prove a more complete inequality of the multiplicity of characteristic.3) Using above theory and method to prove some related conclusions.