Find a mapping from $R^n times n$ to orthogonal matrices in the unit sphere such that distances do not increase.

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Let $O$ be the orthogonal group, let $B = A in R^n times n text lVert A lVert leq 1$.



Find a function $F:R^n times n rightarrow O cap B$, such that $forall_A in R^n times n, Z in B lVert F(A) - Z lVert leq lVert A - Z lVert$.



In other words, I would like to find a mapping from $R^n times n$ to orthogonal matrices in the unit sphere such that distances do not increase.




convex-analysis orthogonality orthogonal-matrices projection




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  • Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
    – user357151
    Jul 30 at 14:48










  • Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
    – mibarg
    Jul 30 at 15:57










  • Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
    – user357151
    Jul 30 at 16:34














up vote
0
down vote

favorite












Let $O$ be the orthogonal group, let $B = A in R^n times n text lVert A lVert leq 1$.



Find a function $F:R^n times n rightarrow O cap B$, such that $forall_A in R^n times n, Z in B lVert F(A) - Z lVert leq lVert A - Z lVert$.



In other words, I would like to find a mapping from $R^n times n$ to orthogonal matrices in the unit sphere such that distances do not increase.




convex-analysis orthogonality orthogonal-matrices projection




share|cite|improve this question



















  • Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
    – user357151
    Jul 30 at 14:48










  • Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
    – mibarg
    Jul 30 at 15:57










  • Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
    – user357151
    Jul 30 at 16:34












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $O$ be the orthogonal group, let $B = A in R^n times n text lVert A lVert leq 1$.



Find a function $F:R^n times n rightarrow O cap B$, such that $forall_A in R^n times n, Z in B lVert F(A) - Z lVert leq lVert A - Z lVert$.



In other words, I would like to find a mapping from $R^n times n$ to orthogonal matrices in the unit sphere such that distances do not increase.




convex-analysis orthogonality orthogonal-matrices projection




share|cite|improve this question











Let $O$ be the orthogonal group, let $B = A in R^n times n text lVert A lVert leq 1$.



Find a function $F:R^n times n rightarrow O cap B$, such that $forall_A in R^n times n, Z in B lVert F(A) - Z lVert leq lVert A - Z lVert$.



In other words, I would like to find a mapping from $R^n times n$ to orthogonal matrices in the unit sphere such that distances do not increase.





convex-analysis orthogonality orthogonal-matrices projection





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 13:43









mibarg

37429




37429











  • Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
    – user357151
    Jul 30 at 14:48










  • Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
    – mibarg
    Jul 30 at 15:57










  • Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
    – user357151
    Jul 30 at 16:34
















  • Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
    – user357151
    Jul 30 at 14:48










  • Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
    – mibarg
    Jul 30 at 15:57










  • Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
    – user357151
    Jul 30 at 16:34















Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
– user357151
Jul 30 at 14:48




Take $Z=Ain Bsetminus O$, then the inequality $|F(A)-Z|le |A-Z|$ cannot hold.
– user357151
Jul 30 at 14:48












Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
– mibarg
Jul 30 at 15:57




Thanks! what if we focus on $Bbb R ^ n times n setminus B$?
– mibarg
Jul 30 at 15:57












Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
– user357151
Jul 30 at 16:34




Does not help if $A$ is very close to a point of $Bsetminus O$. You need to rethink the problem.
– user357151
Jul 30 at 16:34















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