Mixing
Find $A$ and $B$ so that $ operatorname{Tr}(AB) ^{*} =0$
$begingroup$
Here $A,B in M_n(mathbb C) $. I thought about taking diagonal matrices, but I can't figure them out.
Edit: $X^{*}$ denotes the adjugate matrix of $X$ and $A, B neq O_n$
linear-algebra matrices
edited Jan 28 at 9:39
Bernard
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
$endgroup$
add a comment |
$begingroup$
Here $A,B in M_n(mathbb C) $. I thought about taking diagonal matrices, but I can't figure them out.
Edit: $X^{*}$ denotes the adjugate matrix of $X$ and $A, B neq O_n$
linear-algebra matrices
edited Jan 28 at 9:39
Bernard
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
$endgroup$
$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
add a comment |
$begingroup$
Here $A,B in M_n(mathbb C) $. I thought about taking diagonal matrices, but I can't figure them out.
Edit: $X^{*}$ denotes the adjugate matrix of $X$ and $A, B neq O_n$
linear-algebra matrices
edited Jan 28 at 9:39
Bernard
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
$endgroup$
Here $A,B in M_n(mathbb C) $. I thought about taking diagonal matrices, but I can't figure them out.
Edit: $X^{*}$ denotes the adjugate matrix of $X$ and $A, B neq O_n$
linear-algebra matrices
linear-algebra matrices
edited Jan 28 at 9:39
Bernard
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
edited Jan 28 at 9:39
Bernard
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
edited Jan 28 at 9:39
Bernard
123k741117
edited Jan 28 at 9:39
Bernard
123k741117
edited Jan 28 at 9:39
Bernard
123k741117
123k741117
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 6:16
JustAnAmateurJustAnAmateur
1096
1096
$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
add a comment |
$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
add a comment |
1 Answer
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$begingroup$
If all the even columns of $A$ are $0$ and all the odd rows of $B$ are $0$ then $AB=0$ so $Tr(AB)^{*}=0$. For example, take $a_{ij}=0$ when $j$ is even, $1$ when $j$ is odd and $b_{ij}=1$ when $i$ is even, $0$ when $i$ is odd.
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
If all the even columns of $A$ are $0$ and all the odd rows of $B$ are $0$ then $AB=0$ so $Tr(AB)^{*}=0$. For example, take $a_{ij}=0$ when $j$ is even, $1$ when $j$ is odd and $b_{ij}=1$ when $i$ is even, $0$ when $i$ is odd.
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
$endgroup$
add a comment |
$begingroup$
If all the even columns of $A$ are $0$ and all the odd rows of $B$ are $0$ then $AB=0$ so $Tr(AB)^{*}=0$. For example, take $a_{ij}=0$ when $j$ is even, $1$ when $j$ is odd and $b_{ij}=1$ when $i$ is even, $0$ when $i$ is odd.
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
$endgroup$
add a comment |
$begingroup$
If all the even columns of $A$ are $0$ and all the odd rows of $B$ are $0$ then $AB=0$ so $Tr(AB)^{*}=0$. For example, take $a_{ij}=0$ when $j$ is even, $1$ when $j$ is odd and $b_{ij}=1$ when $i$ is even, $0$ when $i$ is odd.
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
$endgroup$
If all the even columns of $A$ are $0$ and all the odd rows of $B$ are $0$ then $AB=0$ so $Tr(AB)^{*}=0$. For example, take $a_{ij}=0$ when $j$ is even, $1$ when $j$ is odd and $b_{ij}=1$ when $i$ is even, $0$ when $i$ is odd.
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
answered Jan 28 at 6:33
Kavi Rama MurthyKavi Rama Murthy
70.4k53170
70.4k53170
add a comment |
add a comment |
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$begingroup$
If you are just looking for examples, you can take $A=B=0$ or any $A$, $B$ such that $AB=0$.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 6:21
$begingroup$
What about $A in M_n(mathbb C)$ arbitrary and $B=0$ ?
$endgroup$
– Fred
Jan 28 at 6:21
$begingroup$
I forgot to mention, both $A$ and $B$ can't be the null matrix.
$endgroup$
– JustAnAmateur
Jan 28 at 6:23
$begingroup$
Surely you mean $Tr((AB)^*)$, not $Tr(AB)^*$...
$endgroup$
– Arturo Magidin
Jan 28 at 6:38
$begingroup$
Also, please include your actual question in the body of the post, not just in the title.
$endgroup$
– Arturo Magidin
Jan 28 at 6:38