A linear algebra: A is a matrix of order n, R (a) = R

A linear algebra: A is a matrix of order n, R (a) = R

There is an invertible matrix P such that PAP ^ (- 1) = B
Where B is a partitioned matrix, and the R * r submatrix B in its upper left corner_ 11 reversible, the other three are 0
Construct M0 = B + C, where C is a partitioned matrix, and its lower right corner is the identity matrix e of (N-R) * (N-R)_ (N-R), the other three are 0
We construct MI, I = 1,..., N-R as follows:
Mi is a diagonal matrix, its diagonal elements are all 1, but there is one exception: the n-i + 1 element is 0
Obviously, B = M0 * M1 *... * m (N-R), where M0 is reversible, R (MI) = n-1, I = 1,..., n-r
So a = P ^ (- 1) BP
= P^(-1)M0*M1*...*M(n-r)P
= D1*D2*.*D(n-r),
Where, D1 = P ^ (- 1) M0 * M1,
Di = Mi,i = 2,...,n-r-1,
D(n-r)=M(n-r)*P,
Is the product of N-R matrices of order n with rank n-1