If matrix $A$ is unitary and $B^2=A$, is $B$ necessarily unitary? The Next CEO of Stack OverflowIf matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitarySquare root of unitary matrixA unitary matrix taking a real matrix to another real matrix, is it an orthogonal matrix?Condition of a unitary matrix“Generalized Unitary Matrix”Decomposition of a unitary matrixGetting a Unitary matrix and the corresponding triangular matrixPositive definite matrix and unitary matrixPolar decomposition and unitary matrixa unitary relation between a matrix and its transposeDecomposition of unitary matrix into phase and special unitarySquare root of unitary matrix
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If matrix $A$ is unitary and $B^2=A$, is $B$ necessarily unitary?
The Next CEO of Stack OverflowIf matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitarySquare root of unitary matrixA unitary matrix taking a real matrix to another real matrix, is it an orthogonal matrix?Condition of a unitary matrix“Generalized Unitary Matrix”Decomposition of a unitary matrixGetting a Unitary matrix and the corresponding triangular matrixPositive definite matrix and unitary matrixPolar decomposition and unitary matrixa unitary relation between a matrix and its transposeDecomposition of unitary matrix into phase and special unitarySquare root of unitary matrix
$begingroup$
If matrix $A$ is unitary and the matrix $B$ satisfies $B^2=A$, is $B$ necessarily unitary?
linear-algebra matrices
edited Mar 26 at 13:21
Micapps
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
$begingroup$
If matrix $A$ is unitary and the matrix $B$ satisfies $B^2=A$, is $B$ necessarily unitary?
linear-algebra matrices
edited Mar 26 at 13:21
Micapps
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
1
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
3
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago
add a comment |
$begingroup$
If matrix $A$ is unitary and the matrix $B$ satisfies $B^2=A$, is $B$ necessarily unitary?
linear-algebra matrices
edited Mar 26 at 13:21
Micapps
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
If matrix $A$ is unitary and the matrix $B$ satisfies $B^2=A$, is $B$ necessarily unitary?
linear-algebra matrices
linear-algebra matrices
edited Mar 26 at 13:21
Micapps
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 26 at 13:21
Micapps
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 26 at 13:21
Micapps
1,15939
edited Mar 26 at 13:21
Micapps
1,15939
edited Mar 26 at 13:21
Micapps
1,15939
1,15939
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
asked Mar 26 at 13:02
Denny ShenDenny Shen
82
82
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Denny Shen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
3
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago
add a comment |
1
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
3
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago
1
1
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
3
3
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let $Ain U_n$.
$textbfProposition$: Case 1. $A$ has $n$ distinct eigenvalues. Then $A$ admits $2^n$ square roots and each of them is unitary.
Case 2. $A$ admits at least one multiple eigenvalue. Then $A$ admits an infinity of square roots that are not unitary and an infinity of square roots that are unitary.
answered 2 days ago
loup blancloup blanc
24.1k21851
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Let $Ain U_n$.
$textbfProposition$: Case 1. $A$ has $n$ distinct eigenvalues. Then $A$ admits $2^n$ square roots and each of them is unitary.
Case 2. $A$ admits at least one multiple eigenvalue. Then $A$ admits an infinity of square roots that are not unitary and an infinity of square roots that are unitary.
answered 2 days ago
loup blancloup blanc
24.1k21851
$endgroup$
add a comment |
$begingroup$
Let $Ain U_n$.
$textbfProposition$: Case 1. $A$ has $n$ distinct eigenvalues. Then $A$ admits $2^n$ square roots and each of them is unitary.
Case 2. $A$ admits at least one multiple eigenvalue. Then $A$ admits an infinity of square roots that are not unitary and an infinity of square roots that are unitary.
answered 2 days ago
loup blancloup blanc
24.1k21851
$endgroup$
add a comment |
$begingroup$
Let $Ain U_n$.
$textbfProposition$: Case 1. $A$ has $n$ distinct eigenvalues. Then $A$ admits $2^n$ square roots and each of them is unitary.
Case 2. $A$ admits at least one multiple eigenvalue. Then $A$ admits an infinity of square roots that are not unitary and an infinity of square roots that are unitary.
answered 2 days ago
loup blancloup blanc
24.1k21851
$endgroup$
Let $Ain U_n$.
$textbfProposition$: Case 1. $A$ has $n$ distinct eigenvalues. Then $A$ admits $2^n$ square roots and each of them is unitary.
Case 2. $A$ admits at least one multiple eigenvalue. Then $A$ admits an infinity of square roots that are not unitary and an infinity of square roots that are unitary.
answered 2 days ago
loup blancloup blanc
24.1k21851
answered 2 days ago
loup blancloup blanc
24.1k21851
answered 2 days ago
loup blancloup blanc
24.1k21851
answered 2 days ago
loup blancloup blanc
24.1k21851
24.1k21851
add a comment |
add a comment |
Denny Shen is a new contributor. Be nice, and check out our Code of Conduct.
Denny Shen is a new contributor. Be nice, and check out our Code of Conduct.
Denny Shen is a new contributor. Be nice, and check out our Code of Conduct.
Denny Shen is a new contributor. Be nice, and check out our Code of Conduct.
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1
$begingroup$
I'm not sure what you mean by "define the following statement". Do you mean to ask: "Suppose $A$ is unitary and $B$ satisfies $B^2=A$. Is $B$ necessarily unitary?"
$endgroup$
– Micapps
Mar 26 at 13:05
$begingroup$
Yes, you are right.
$endgroup$
– Denny Shen
Mar 26 at 13:09
3
$begingroup$
Hint: take $A$ to be the identity matrix.
$endgroup$
– Andreas Caranti
Mar 26 at 13:11
$begingroup$
math.stackexchange.com/questions/2717389/…
$endgroup$
– daw
2 days ago