Ab=0 iff A^+b=0, when A is symmetric.

Let \( A' \) be transpose of \( A \), \( A^{+} \) be MP-inverse of \( A \).

Prove that if \( A=A' \), then \(A b=0 \) if and only if \( A^+b=0 \).

Proof:

\( A b=0 \) implies that \( b=(I-A^+A)q \) for some \( q \).  Then,

\(
A^+ b \\
=A^+(I-A^+A)q \\
=A^+(I-A'^+A')q \\
=A^+(I-(AA^+)')q \\
=A^+(I-AA^+)q \\
=A^+q-A^+AA^+q \\
=A^+q-A^+q \\
=0
\)

Because \( A^{++}=A \), the proof is completed.