The well-known matrix inequality (AB)~(-1)≤A~(-1)B~(-1) involving Hadamard product of positive definite matrices is refined to(AB)~(-1)≤diag((A~(-1)(α)~(-1)B(α))~(-1),(A(α′)B~(-1)(α′)~(-1))~(-1)),≤diag(A~(-1)(α)B(α)~(-1),A(α′)~(-1)B~(-1)(α′))≤A~(-1)B~(-1),where A(α) is the leading principal submatrix and α′ is the complementary sequence of α.

With the development of computer architecture and the introduction of cache, blocking has been the main method to optimize performance in matrix computing, and the effect of leading dimension becomes important to blocking algorithms performance.