Proving the Spectral Theorem in Matrix Algebra

[SNOOTCHIEBOOCHEE]

Homework Statement

Prove that for any square matrix A, ker A= (im A*)^$$\bot$$

Homework Equations

A* = A conjugate transpose

The Attempt at a Solution

so we want to show that all A that satisfies Ax=0 for some x, that set is equal to (im A*)^$$\bot$$

But i really don't know what image of this could possibly be... we arent given a function.

 

[jimmypoopins]

But i really don't know what image of this could possibly be... we arent given a function.

it holds for any function.

prove it for any function (you should be able to)

 

[SNOOTCHIEBOOCHEE]

So the image of the null space of A* is equal to the null space of A.

Im thinking this proof leads me in the direction that I am A* = A. then we are done. but i don't know how to show that.