What is the missing piece in proving A=0 when A(A*)=0?

Click For Summary

Homework Help Overview

The discussion revolves around proving a property related to matrices, specifically addressing the implications of the product of a matrix and its conjugate transpose being zero. The original poster questions the conditions under which A(A*) = 0 leads to A = 0.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the implications of the equation A(A*) = 0 and expresses difficulty in concluding that A must be zero. Participants explore the significance of diagonal elements in the context of inner products and their relationship to the zero condition.

Discussion Status

Some participants have provided insights that suggest focusing on the diagonal elements of the product (A*)A, leading to a realization that zero diagonal entries may imply A = 0. However, the discussion remains open regarding the implications of A(A*).

Contextual Notes

There is an exploration of the conditions under which the inner product of vectors results in zero, as well as the implications of these conditions for the matrix A. The discussion does not resolve the broader question of A(A*) = 0.

mathboy
Messages
182
Reaction score
0
Let A be a nxn matrix. Prove that if (A*)A=0 then A=0. What if A(A*) = 0?

A* is the conjugate transpose of A. When I write out the expansion formula, I cannot conclude that every entry of A is zero. What am I missing?
 
Physics news on Phys.org
Concentrate on the diagonal elements of (A*)A. Each one is the inner product of a row of A* with the corresponding column of A. I.e. it's the inner product of a vector with it's conjugate transpose. Under what conditions can that be zero?
 
Oh, that was a clever idea. I got it now. So it turns out the weaker condition of (A*)A having zero diagaonal entries is enough to conclude that A=0. And the same is true of A(A*).
 
Bingo. You've got it.
 

Similar threads

Undergrad Erroneously finding discrepancy in transpose rule
  • · Replies 13 ·
Replies
13
Views
2K
JavaScript Need of transposed[i] = []; when we already have var transposed = []
  • · Replies 1 ·
Replies
1
Views
2K
Product of two boosts and then two inverse boosts is rotation matrix
  • · Replies 13 ·
Replies
13
Views
2K
How Can We Prove the Conjugate Transpose Property of Complex Matrices?
  • · Replies 5 ·
Replies
5
Views
2K
Prove the identity matrix is unique
  • · Replies 69 ·
3
Replies
69
Views
11K
How Do Hadamard Matrices Relate to Identity Matrices?
  • · Replies 2 ·
Replies
2
Views
2K
A gardener collected 17 apples...
  • · Replies 32 ·
2
Replies
32
Views
3K
Solving (v+c)^3-3vc(v+c)-(v^3+c^3)=0: What am I Missing?
  • · Replies 12 ·
Replies
12
Views
2K
Graduate Trace of the inverse of matrix products
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Uncertainties in the proof of Proposition 4.4.2 in Hawking and Ellis
  • · Replies 2 ·
Replies
2
Views
1K