How to prove that for any A matrix ϱ(A*A)=ϱ(A) is true
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Prove that for any matrix $A$ the statement $ϱ(A^*A)=ϱ(A)$ is true, where $A^*$ is the conjugate transpose of $A$.
linear-algebra
edited Nov 13 at 20:13
martini
70k45990
asked Nov 13 at 19:41
nyaki
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up vote
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Prove that for any matrix $A$ the statement $ϱ(A^*A)=ϱ(A)$ is true, where $A^*$ is the conjugate transpose of $A$.
linear-algebra
edited Nov 13 at 20:13
martini
70k45990
asked Nov 13 at 19:41
nyaki
1
what is $rho$?
– David Hill
Nov 13 at 19:47
The rank of the matrix.
– nyaki
Nov 13 at 19:53
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Prove that for any matrix $A$ the statement $ϱ(A^*A)=ϱ(A)$ is true, where $A^*$ is the conjugate transpose of $A$.
linear-algebra
edited Nov 13 at 20:13
martini
70k45990
asked Nov 13 at 19:41
nyaki
1
Prove that for any matrix $A$ the statement $ϱ(A^*A)=ϱ(A)$ is true, where $A^*$ is the conjugate transpose of $A$.
linear-algebra
linear-algebra
edited Nov 13 at 20:13
martini
70k45990
asked Nov 13 at 19:41
nyaki
1
edited Nov 13 at 20:13
martini
70k45990
asked Nov 13 at 19:41
nyaki
1
edited Nov 13 at 20:13
martini
70k45990
edited Nov 13 at 20:13
martini
70k45990
edited Nov 13 at 20:13
martini
70k45990
70k45990
asked Nov 13 at 19:41
nyaki
1
asked Nov 13 at 19:41
nyaki
1
asked Nov 13 at 19:41
nyaki
1
1
what is $rho$?
– David Hill
Nov 13 at 19:47
The rank of the matrix.
– nyaki
Nov 13 at 19:53
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46
add a comment |
what is $rho$?
– David Hill
Nov 13 at 19:47
The rank of the matrix.
– nyaki
Nov 13 at 19:53
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46
what is $rho$?
– David Hill
Nov 13 at 19:47
what is $rho$?
– David Hill
Nov 13 at 19:47
The rank of the matrix.
– nyaki
Nov 13 at 19:53
The rank of the matrix.
– nyaki
Nov 13 at 19:53
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46
add a comment |
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what is $rho$?
– David Hill
Nov 13 at 19:47
The rank of the matrix.
– nyaki
Nov 13 at 19:53
Weaker, but easier: Since $det(A^*A)=|det(A)|^2$, $A$ is invertible exactly when $A^*A$ is invertible. Which, btw, is all you actually care about.
– David Hill
Nov 14 at 4:46