5

Is there any proof that a MPI of symetric matrix is symmetric matrix? Basically I need that Moore-Penroses invers of positive semidefinite matrix is positive semidefinite. I can show that x^T(A+)x >=0. But A+ also need to be symmetric for positive semidefinite.

Thank you for your help.

Glass12
  • 83
  • 1
    is this correct proof od this question? A is symmetric so A=A^T, and denote MPI A+ with B.

    Then: A=A^T (ABA)=(ABA)^T
    A=(A)^T(B)^T(A)^T, since A=A^T
    A=A (B)^T A, so I wrote this with some other B, but B from Moore-Penroses invers is unique, so is this proof that B=B^T?

    Thank you.

     – Glass12 Nov 08 '15 at 01:10

1 Answers1

7

Let $A$ be a symmetric matrix and let $B$ be its pseudoinverse. By definition, this means that $B$ is the unique matrix satisfying:

  1. $ABA=A$
  2. $BAB=B$
  3. $AB$ and $BA$ are symmetric

Taking transposes of these equations implies that:

  1. $AB^TA=A$
  2. $B^TAB^T=B^T$
  3. $B^TA$ and $AB^T$ are symmetric

Thus $B^T$ is also a pseudoinverse for $A$. By uniqueness, it follows that $B$ is symmetric.

pre-kidney
  • 30,223