If $A=AA^{\top}$, show that $A^2=A$

Question

I've been working trying to understand the following question:

Let n be a positive integer, let $F$ be a field, and let $A \in \mathrm{Mat}(n,F)$ satisfy the condition $A=AA^{\top}$. Show that $A^2=A$.

I haven't made much progress since my knowledge is pretty basic but I ran across this link and was wondering if this example was essentially the same?

Example

If $A=A A^T$, can you express $A^T$? – Peter Franek Jun 30 '14 at 17:10 — Peter Franek, Jun 30 '14 at 17:10

score 20 · Accepted Answer · answered Jun 30 '14 at 17:11

20

Since $A^T = (AA^T)^T = AA^T = A$, You have $A^2 = AA = AA^T = A$

answered Jun 30 '14 at 17:11

Ant

21,522

I follow the A^T til =A, the first part.Then A^2=AA but how does AA=AA^T? – cele Jun 30 '14 at 17:48
1

Use the fact $A^T = A$ to substitute $A^T$ for the second $A$ in $AA$. – David K Jun 30 '14 at 17:54
oh, ok. I figured it was something plain as day. Thanks – cele Jun 30 '14 at 17:59

score 6 · Answer 2 · answered Jun 30 '14 at 17:11

6

get transpose from $A=AA^T$ thus we have $A^T=AA^T$ and thus we have $A=A^T$ and it proved that $A=A^2$

answered Jun 30 '14 at 17:11

callculus42 · Answer 3 · 2014-07-01T00:19:24.353

-1

$AA^T=A$

$A$ from the LHS to the RHS

$A^T=AA^{-1} \Rightarrow A^T=I\Rightarrow A=I$

$I^2=I \Rightarrow A^2=A$

Edit: It works only, if $det \ A\neq 0$

edited Jul 01 '14 at 00:19

answered Jun 30 '14 at 17:39

callculus42

31,012

1

Consider $A=0$, which has no inverse. – abnry Jun 30 '14 at 17:41
Thank you for your hint. I didn´t think of this case. – callculus42 Jun 30 '14 at 17:46
And $A^2 = A$ in the special case $A = 0$ too, although not via the exact same reasoning. – David K Jun 30 '14 at 17:56

If $A=AA^{\top}$, show that $A^2=A$

3 Answers3