Mixing
Can a Hermitian Matrix be Decomposed into a Sum of Unitary Matricies?
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Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form?
$$ A = sum_{i} a_i U$$
Where $U$ is unitary.
Intuitively, because you have a nice SVD: $A = U Sigma U^{dagger}$, I would expect to see that this is always possible with the coefficients somehow related to the eigenvalues, but I am not entirely sure.
linear-algebra matrices
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
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add a comment |
$begingroup$
Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form?
$$ A = sum_{i} a_i U$$
Where $U$ is unitary.
Intuitively, because you have a nice SVD: $A = U Sigma U^{dagger}$, I would expect to see that this is always possible with the coefficients somehow related to the eigenvalues, but I am not entirely sure.
linear-algebra matrices
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
$endgroup$
add a comment |
$begingroup$
Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form?
$$ A = sum_{i} a_i U$$
Where $U$ is unitary.
Intuitively, because you have a nice SVD: $A = U Sigma U^{dagger}$, I would expect to see that this is always possible with the coefficients somehow related to the eigenvalues, but I am not entirely sure.
linear-algebra matrices
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
$endgroup$
Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form?
$$ A = sum_{i} a_i U$$
Where $U$ is unitary.
Intuitively, because you have a nice SVD: $A = U Sigma U^{dagger}$, I would expect to see that this is always possible with the coefficients somehow related to the eigenvalues, but I am not entirely sure.
linear-algebra matrices
linear-algebra matrices
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
asked Apr 24 '16 at 3:21
KF GaussKF Gauss
1627
1627
add a comment |
add a comment |
1 Answer
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$begingroup$
I was too restrictive when I wrote this question and missed this result:
Every matrix can be written as a sum of unitary matrices?
Turns out every complex square matrix can be written as a linear sum of at most two unitary matricies.
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I was too restrictive when I wrote this question and missed this result:
Every matrix can be written as a sum of unitary matrices?
Turns out every complex square matrix can be written as a linear sum of at most two unitary matricies.
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
$endgroup$
add a comment |
$begingroup$
I was too restrictive when I wrote this question and missed this result:
Every matrix can be written as a sum of unitary matrices?
Turns out every complex square matrix can be written as a linear sum of at most two unitary matricies.
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
$endgroup$
add a comment |
$begingroup$
I was too restrictive when I wrote this question and missed this result:
Every matrix can be written as a sum of unitary matrices?
Turns out every complex square matrix can be written as a linear sum of at most two unitary matricies.
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
$endgroup$
I was too restrictive when I wrote this question and missed this result:
Every matrix can be written as a sum of unitary matrices?
Turns out every complex square matrix can be written as a linear sum of at most two unitary matricies.
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
edited Jan 28 at 0:26
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
answered Apr 24 '16 at 3:31
KF GaussKF Gauss
1627
1627
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