Is the determinant of a complex matrix the complex conjugate of the determinant of it's complex conjugate...
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Apologies for the confusing title.
Suppose we have some square matrix $A$ with complex entries and it's conjugate matrix $bar{A}$ whose entries are the complex conjugate of those in $A$.
Is it true that the determinant of one of them is the complex conjugate of the determinant of the other? It seems simple for small matrices but I don't know if it's true in general (or is there a transpose in there somewhere?). In other words does:
$$overline{left(|A|right)}={|overline{A}|}$$
matrices complex-numbers
asked Dec 3 at 4:26
R. Rankin
305213
add a comment |
up vote
1
down vote
favorite
Apologies for the confusing title.
Suppose we have some square matrix $A$ with complex entries and it's conjugate matrix $bar{A}$ whose entries are the complex conjugate of those in $A$.
Is it true that the determinant of one of them is the complex conjugate of the determinant of the other? It seems simple for small matrices but I don't know if it's true in general (or is there a transpose in there somewhere?). In other words does:
$$overline{left(|A|right)}={|overline{A}|}$$
matrices complex-numbers
asked Dec 3 at 4:26
R. Rankin
305213
You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Apologies for the confusing title.
Suppose we have some square matrix $A$ with complex entries and it's conjugate matrix $bar{A}$ whose entries are the complex conjugate of those in $A$.
Is it true that the determinant of one of them is the complex conjugate of the determinant of the other? It seems simple for small matrices but I don't know if it's true in general (or is there a transpose in there somewhere?). In other words does:
$$overline{left(|A|right)}={|overline{A}|}$$
matrices complex-numbers
asked Dec 3 at 4:26
R. Rankin
305213
Apologies for the confusing title.
Suppose we have some square matrix $A$ with complex entries and it's conjugate matrix $bar{A}$ whose entries are the complex conjugate of those in $A$.
Is it true that the determinant of one of them is the complex conjugate of the determinant of the other? It seems simple for small matrices but I don't know if it's true in general (or is there a transpose in there somewhere?). In other words does:
$$overline{left(|A|right)}={|overline{A}|}$$
matrices complex-numbers
matrices complex-numbers
asked Dec 3 at 4:26
R. Rankin
305213
asked Dec 3 at 4:26
R. Rankin
305213
edited Dec 3 at 21:41
asked Dec 3 at 4:26
R. Rankin
305213
asked Dec 3 at 4:26
R. Rankin
305213
asked Dec 3 at 4:26
R. Rankin
305213
305213
You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03
add a comment |
You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03
You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03
You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
For $z,w in mathbb{C}$, we have $overline{zw} = overline{z} overline{w}$ and $overline{z+w} = overline{z} + overline{w}$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $overline{det A} = det overline{A}$ holds.
Regarding your last sentence, note also that transposing a matrix does not change its determinant.
answered Dec 3 at 4:31
angryavian
37.8k13180
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
For $z,w in mathbb{C}$, we have $overline{zw} = overline{z} overline{w}$ and $overline{z+w} = overline{z} + overline{w}$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $overline{det A} = det overline{A}$ holds.
Regarding your last sentence, note also that transposing a matrix does not change its determinant.
answered Dec 3 at 4:31
angryavian
37.8k13180
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
add a comment |
up vote
2
down vote
accepted
For $z,w in mathbb{C}$, we have $overline{zw} = overline{z} overline{w}$ and $overline{z+w} = overline{z} + overline{w}$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $overline{det A} = det overline{A}$ holds.
Regarding your last sentence, note also that transposing a matrix does not change its determinant.
answered Dec 3 at 4:31
angryavian
37.8k13180
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
For $z,w in mathbb{C}$, we have $overline{zw} = overline{z} overline{w}$ and $overline{z+w} = overline{z} + overline{w}$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $overline{det A} = det overline{A}$ holds.
Regarding your last sentence, note also that transposing a matrix does not change its determinant.
answered Dec 3 at 4:31
angryavian
37.8k13180
For $z,w in mathbb{C}$, we have $overline{zw} = overline{z} overline{w}$ and $overline{z+w} = overline{z} + overline{w}$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $overline{det A} = det overline{A}$ holds.
Regarding your last sentence, note also that transposing a matrix does not change its determinant.
answered Dec 3 at 4:31
angryavian
37.8k13180
answered Dec 3 at 4:31
angryavian
37.8k13180
answered Dec 3 at 4:31
angryavian
37.8k13180
answered Dec 3 at 4:31
angryavian
37.8k13180
37.8k13180
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
add a comment |
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
Thanks, exactly what I needed, I'll accept after the allowed time has passed
– R. Rankin
Dec 3 at 4:34
add a comment |
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You can delete your last sentence if you are satisfied with the edited formula.
– user376343
Dec 3 at 12:03