Do orthonormal matrices have a geometric interpretation or contextual importance? [closed]












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I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and perhaps another set of matrices, but so what ?










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closed as off-topic by José Carlos Santos, RRL, Did, Alexander Gruber Jan 9 at 0:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, RRL, Did, Alexander Gruber

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 3




    $begingroup$
    The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
    $endgroup$
    – jgon
    Jan 1 at 17:45








  • 1




    $begingroup$
    What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
    $endgroup$
    – José Carlos Santos
    Jan 1 at 17:45










  • $begingroup$
    I mean for instance that you can have matrices that don't have 1s or 0s.
    $endgroup$
    – Kid Cudi
    Jan 1 at 17:47










  • $begingroup$
    If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
    $endgroup$
    – littleO
    Jan 1 at 17:47










  • $begingroup$
    Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
    $endgroup$
    – saulspatz
    Jan 1 at 17:56
















0












$begingroup$


I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and perhaps another set of matrices, but so what ?










share|cite|improve this question











$endgroup$



closed as off-topic by José Carlos Santos, RRL, Did, Alexander Gruber Jan 9 at 0:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, RRL, Did, Alexander Gruber

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 3




    $begingroup$
    The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
    $endgroup$
    – jgon
    Jan 1 at 17:45








  • 1




    $begingroup$
    What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
    $endgroup$
    – José Carlos Santos
    Jan 1 at 17:45










  • $begingroup$
    I mean for instance that you can have matrices that don't have 1s or 0s.
    $endgroup$
    – Kid Cudi
    Jan 1 at 17:47










  • $begingroup$
    If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
    $endgroup$
    – littleO
    Jan 1 at 17:47










  • $begingroup$
    Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
    $endgroup$
    – saulspatz
    Jan 1 at 17:56














0












0








0


2



$begingroup$


I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and perhaps another set of matrices, but so what ?










share|cite|improve this question











$endgroup$




I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and perhaps another set of matrices, but so what ?







linear-algebra orthonormal orthogonal-matrices






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edited Jan 1 at 18:35









David G. Stork

11k41432




11k41432










asked Jan 1 at 17:42









Kid CudiKid Cudi

456




456




closed as off-topic by José Carlos Santos, RRL, Did, Alexander Gruber Jan 9 at 0:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, RRL, Did, Alexander Gruber

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by José Carlos Santos, RRL, Did, Alexander Gruber Jan 9 at 0:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, RRL, Did, Alexander Gruber

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 3




    $begingroup$
    The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
    $endgroup$
    – jgon
    Jan 1 at 17:45








  • 1




    $begingroup$
    What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
    $endgroup$
    – José Carlos Santos
    Jan 1 at 17:45










  • $begingroup$
    I mean for instance that you can have matrices that don't have 1s or 0s.
    $endgroup$
    – Kid Cudi
    Jan 1 at 17:47










  • $begingroup$
    If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
    $endgroup$
    – littleO
    Jan 1 at 17:47










  • $begingroup$
    Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
    $endgroup$
    – saulspatz
    Jan 1 at 17:56














  • 3




    $begingroup$
    The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
    $endgroup$
    – jgon
    Jan 1 at 17:45








  • 1




    $begingroup$
    What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
    $endgroup$
    – José Carlos Santos
    Jan 1 at 17:45










  • $begingroup$
    I mean for instance that you can have matrices that don't have 1s or 0s.
    $endgroup$
    – Kid Cudi
    Jan 1 at 17:47










  • $begingroup$
    If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
    $endgroup$
    – littleO
    Jan 1 at 17:47










  • $begingroup$
    Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
    $endgroup$
    – saulspatz
    Jan 1 at 17:56








3




3




$begingroup$
The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
$endgroup$
– jgon
Jan 1 at 17:45






$begingroup$
The geometric significance of orthonormal matrices is that they describe linear transformations that preserve angles between vectors and magnitudes of vectors. Also they're usually called orthogonal matrices.
$endgroup$
– jgon
Jan 1 at 17:45






1




1




$begingroup$
What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
$endgroup$
– José Carlos Santos
Jan 1 at 17:45




$begingroup$
What do you mean by “the axis would be othonormal” and “perhaps another set of matrices”?
$endgroup$
– José Carlos Santos
Jan 1 at 17:45












$begingroup$
I mean for instance that you can have matrices that don't have 1s or 0s.
$endgroup$
– Kid Cudi
Jan 1 at 17:47




$begingroup$
I mean for instance that you can have matrices that don't have 1s or 0s.
$endgroup$
– Kid Cudi
Jan 1 at 17:47












$begingroup$
If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
$endgroup$
– littleO
Jan 1 at 17:47




$begingroup$
If $Q$ is a real $n times n$ matrix with orthonormal columns, then $Q^T$ is the change of basis matrix from the standard basis to the basis consisting of columns of $Q$.
$endgroup$
– littleO
Jan 1 at 17:47












$begingroup$
Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
$endgroup$
– saulspatz
Jan 1 at 17:56




$begingroup$
Rotation matrices are orthogonal matrices whose elements are not 0s and 1s, in general. Do you know how to right down the matrix of a plane rotation?
$endgroup$
– saulspatz
Jan 1 at 17:56










1 Answer
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When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what do you mention under distance of vectors?
    $endgroup$
    – user376343
    Jan 1 at 18:44


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what do you mention under distance of vectors?
    $endgroup$
    – user376343
    Jan 1 at 18:44
















1












$begingroup$

When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what do you mention under distance of vectors?
    $endgroup$
    – user376343
    Jan 1 at 18:44














1












1








1





$begingroup$

When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).






share|cite|improve this answer









$endgroup$



When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 1 at 18:37









DanielWainfleetDanielWainfleet

35.3k31648




35.3k31648












  • $begingroup$
    what do you mention under distance of vectors?
    $endgroup$
    – user376343
    Jan 1 at 18:44


















  • $begingroup$
    what do you mention under distance of vectors?
    $endgroup$
    – user376343
    Jan 1 at 18:44
















$begingroup$
what do you mention under distance of vectors?
$endgroup$
– user376343
Jan 1 at 18:44




$begingroup$
what do you mention under distance of vectors?
$endgroup$
– user376343
Jan 1 at 18:44



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