Question

9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A € Mmxm(R) is the matrix A+ E Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+ AA+ = A+ (c) (AA+)? = AA+ (d) (A+A)T = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-I. The following is true: Any matrix A € Mmxm(R) has a unique pesudo-inverse A+ E Mmxm(R). (69) (note that this matrix does not have (a) (3 points) Find the unique pseudo-inverse of A = an inverse) (b) (1 point) Prove that (AB)+ = B+A+. (c) (1 point) Prove that (A+)+ = A. (d) (1 point) Prove that A+ = A' when A is invertible.

Answer 1

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