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Example text
Proof. Let U be an arbitrary matrix. Then (G + U(I − AG))AA = GAA = A . So, G + U(I − AG) is a minimum norm g-inverse of A for all U. Let G1 be any minimum norm g-inverse of A. Then it is easy to check that G1 = G + U(I − AG), where U = G1 − G. Thus, the class of all minimum norm g-inverse is given by G + U(I − AG), where U is arbitrary. 8. Let A be an m × n matrix of rank r (> 0). Let Udiag(D, 0)V be a singular value decomposition of A, where U and V are unitary and D is a positive definite diagonal matrix of order r×r.
16. Let A be a matrix of index ≤ 1. If λ is a non-null eigenvalue of A with algebraic multiplicity k, then 1/λ is an eigen-value of A# with same algebraic multiplicity. Further, if zero is an eigen-value of A with algebraic multiplicity t, then zero is an eigen-value of A# with the same algebraic multiplicity t. 17. A square matrix is of index not greater than 1 if and only if the algebraic and geometric multiplicities of its zero eigen-value are equal. If A is a non-singular matrix, then it is well known that A−1 is a polynomial in A.
Thus, X is the Drazin inverse of A. 32. Let A be an n × n matrix of index ≤ 1 and (P, Q) be a rank factorization of A. Then A# = P(QP)−2 Q . 5 Moore-Penrose inverse In this section we specialize to vectors and matrices over the field of complex numbers C and use the inner product (x, y) = y x for x, y ∈ Cn . Let Ax = b be a consistent system of linear equations for A ∈ Cm×n and b ∈ Cm . 3, we noticed that, in general, a solution to Ax = b is not unique. Since there are many solutions, we can look for solutions with some optimal property.