3阶实方阵的实正交—对称和分解

Sum decomposition of three order real square matrix

对于任意 3阶实方阵A =(aij) 3× 3,记Δ =(a12 -a2 1) 2 +(a13 -a31) 2 +(a2 3 -a32 ) 2 ,本文证明 :当Δ≤ 4时 ,则存在正交实方阵B与对称实方阵C ,使A =B +C ,且 |B|=1;而当Δ >4时 ,则存在实数α =tΔ (t≥ 1) ,正交实方阵B与对称实方阵C使A =α(B +C) ,且

This article comes to the conclusion that three order reqular intersection real square matries can be decomposed into the product of a regular intersection real square matrix and a real symmetric square matrix.It is proved by the sum decomposition of non symmetric real square matrix that to an arbitrary three order real square matrix A=(a ij ) 3×3 ,recorded as Δ=(a 12 -a 21 ) 2 + (a 13 -a 31 ) 2+(a 23 -a 32 ) 2 ,the regular intersection real square matrix B and the symmetric real square matrix C can exist. When Δ ≤4, we can get A=B+C , just ｜ B ｜=1; When Δ >4,the real number α=tΔ (t≥1) can exist,then we can get A=α(B+C) , just ｜B｜=1.