Extracting the phase of a determinant
up vote
2
down vote
favorite
Is it possible to extract the phase of a determinant without computing the full determinant?
More explictly, given a complex matrix $U$, the determinant can be written in the form
begin{equation}
text{det}(U) = r e^{itheta}.
end{equation}
Is it possible to extract $theta$ from $U$ without computing the full determinant?
Edit : As pointed by Hans Engler below, this can be done if we are happy with more work than computing the original determinant. I am looking for a method that is more efficient than computing the determinant itself.
linear-algebra matrices complex-numbers determinant
asked Nov 21 at 22:05
as2457
10412
add a comment |
up vote
2
down vote
favorite
Is it possible to extract the phase of a determinant without computing the full determinant?
More explictly, given a complex matrix $U$, the determinant can be written in the form
begin{equation}
text{det}(U) = r e^{itheta}.
end{equation}
Is it possible to extract $theta$ from $U$ without computing the full determinant?
Edit : As pointed by Hans Engler below, this can be done if we are happy with more work than computing the original determinant. I am looking for a method that is more efficient than computing the determinant itself.
linear-algebra matrices complex-numbers determinant
asked Nov 21 at 22:05
as2457
10412
1
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Is it possible to extract the phase of a determinant without computing the full determinant?
More explictly, given a complex matrix $U$, the determinant can be written in the form
begin{equation}
text{det}(U) = r e^{itheta}.
end{equation}
Is it possible to extract $theta$ from $U$ without computing the full determinant?
Edit : As pointed by Hans Engler below, this can be done if we are happy with more work than computing the original determinant. I am looking for a method that is more efficient than computing the determinant itself.
linear-algebra matrices complex-numbers determinant
asked Nov 21 at 22:05
as2457
10412
Is it possible to extract the phase of a determinant without computing the full determinant?
More explictly, given a complex matrix $U$, the determinant can be written in the form
begin{equation}
text{det}(U) = r e^{itheta}.
end{equation}
Is it possible to extract $theta$ from $U$ without computing the full determinant?
Edit : As pointed by Hans Engler below, this can be done if we are happy with more work than computing the original determinant. I am looking for a method that is more efficient than computing the determinant itself.
linear-algebra matrices complex-numbers determinant
linear-algebra matrices complex-numbers determinant
asked Nov 21 at 22:05
as2457
10412
asked Nov 21 at 22:05
as2457
10412
edited Nov 22 at 10:58
asked Nov 21 at 22:05
as2457
10412
asked Nov 21 at 22:05
as2457
10412
asked Nov 21 at 22:05
as2457
10412
10412
1
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10
add a comment |
1
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10
1
1
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^ast)^{-1} U$. Then $det W = e^{2i theta}$.
You have found $theta$ without computing $r$.
answered Nov 21 at 22:43
Hans Engler
9,97411836
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
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
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^ast)^{-1} U$. Then $det W = e^{2i theta}$.
You have found $theta$ without computing $r$.
answered Nov 21 at 22:43
Hans Engler
9,97411836
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
add a comment |
up vote
2
down vote
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^ast)^{-1} U$. Then $det W = e^{2i theta}$.
You have found $theta$ without computing $r$.
answered Nov 21 at 22:43
Hans Engler
9,97411836
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
add a comment |
up vote
2
down vote
up vote
2
down vote
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^ast)^{-1} U$. Then $det W = e^{2i theta}$.
You have found $theta$ without computing $r$.
answered Nov 21 at 22:43
Hans Engler
9,97411836
Yes, it's possible. Here is a way that is unfortunately a lot more work. Compute $W = (U^ast)^{-1} U$. Then $det W = e^{2i theta}$.
You have found $theta$ without computing $r$.
answered Nov 21 at 22:43
Hans Engler
9,97411836
answered Nov 21 at 22:43
Hans Engler
9,97411836
answered Nov 21 at 22:43
Hans Engler
9,97411836
answered Nov 21 at 22:43
Hans Engler
9,97411836
9,97411836
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
add a comment |
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
Thanks for your answer. As you've guessed, I'm also interested in a method that is less work than the determinant itself. I will edit the question to add this condition.
– as2457
Nov 22 at 10:56
add a comment |
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1
Since computing a determinant involves both addition and multiplication this sounds extremely unlikely. But of course that would make it only more interesting when it turns out the answer is in fact yes. So against better judgement I'll wait here for some of the other posters to perform a miracle.
– Vincent
Nov 21 at 22:10