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 ?
linear-algebra orthonormal orthogonal-matrices
edited Jan 1 at 18:35
David G. Stork
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
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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.
|
show 1 more comment
$begingroup$
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 ?
linear-algebra orthonormal orthogonal-matrices
edited Jan 1 at 18:35
David G. Stork
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
$endgroup$
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
|
show 1 more comment
$begingroup$
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 ?
linear-algebra orthonormal orthogonal-matrices
edited Jan 1 at 18:35
David G. Stork
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
$endgroup$
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
linear-algebra orthonormal orthogonal-matrices
edited Jan 1 at 18:35
David G. Stork
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
edited Jan 1 at 18:35
David G. Stork
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
edited Jan 1 at 18:35
David G. Stork
11k41432
edited Jan 1 at 18:35
David G. Stork
11k41432
edited Jan 1 at 18:35
David G. Stork
11k41432
11k41432
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
asked Jan 1 at 17:42
Kid CudiKid Cudi
456
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
|
show 1 more comment
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
|
show 1 more comment
1 Answer
1
active
oldest
votes
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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).
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
$begingroup$
what do you mention under distance of vectors?
$endgroup$
– user376343
Jan 1 at 18:44
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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).
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
$begingroup$
what do you mention under distance of vectors?
$endgroup$
– user376343
Jan 1 at 18:44
add a comment |
$begingroup$
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).
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
$begingroup$
what do you mention under distance of vectors?
$endgroup$
– user376343
Jan 1 at 18:44
add a comment |
$begingroup$
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).
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
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).
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
answered Jan 1 at 18:37
DanielWainfleetDanielWainfleet
35.3k31648
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
add a comment |
$begingroup$
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
add a comment |
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