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generalized inverse of A is equal to A' if A'A is idempotent?
A generalized inverse of a matrix A is a matrix A' that satisfies the conditions A'A = A' and AA'A = A.
Proving that A' is the generalized inverse of A when A'A is idempotent ensures that A' can be used to solve equations involving A, which is crucial in many fields of science and engineering.
A matrix A is idempotent if A*A = A, meaning that multiplying the matrix by itself once results in the same matrix.
To prove that A' is the generalized inverse of A when A'A is idempotent, we must show that A'A = A' and AA'A = A. This can be done by multiplying A' and A, and then using the properties of idempotent matrices to simplify the equations.
No, A' cannot be the generalized inverse of A if A'A is not idempotent. In order for A' to be the generalized inverse, it must satisfy the conditions A'A = A' and AA'A = A, which can only be true if A'A is idempotent.