Alternate inner products on Euclidean space?
$begingroup$
After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up to constant scaling).
But it seems that there are a whole bunch of alternative inner products in $mathbb{R}^2$ with nonzero cross-terms between basis vectors, for example, $langle (a, b)^intercal, (x, y)^intercal rangle = ax + by + 0.5(ay + bx)$. Unless I've made a mistake, this satisfies symmetry, linearity, and positive-definiteness.
Is there a sense in which the dot product is the canonical inner product on Euclidean space? Or do we just pick it because the implied norm matches our notion of distance?
linear-algebra inner-product-space
edited 11 hours ago
Björn Friedrich
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
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rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up to constant scaling).
But it seems that there are a whole bunch of alternative inner products in $mathbb{R}^2$ with nonzero cross-terms between basis vectors, for example, $langle (a, b)^intercal, (x, y)^intercal rangle = ax + by + 0.5(ay + bx)$. Unless I've made a mistake, this satisfies symmetry, linearity, and positive-definiteness.
Is there a sense in which the dot product is the canonical inner product on Euclidean space? Or do we just pick it because the implied norm matches our notion of distance?
linear-algebra inner-product-space
edited 11 hours ago
Björn Friedrich
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
1
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago
add a comment |
$begingroup$
After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up to constant scaling).
But it seems that there are a whole bunch of alternative inner products in $mathbb{R}^2$ with nonzero cross-terms between basis vectors, for example, $langle (a, b)^intercal, (x, y)^intercal rangle = ax + by + 0.5(ay + bx)$. Unless I've made a mistake, this satisfies symmetry, linearity, and positive-definiteness.
Is there a sense in which the dot product is the canonical inner product on Euclidean space? Or do we just pick it because the implied norm matches our notion of distance?
linear-algebra inner-product-space
edited 11 hours ago
Björn Friedrich
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up to constant scaling).
But it seems that there are a whole bunch of alternative inner products in $mathbb{R}^2$ with nonzero cross-terms between basis vectors, for example, $langle (a, b)^intercal, (x, y)^intercal rangle = ax + by + 0.5(ay + bx)$. Unless I've made a mistake, this satisfies symmetry, linearity, and positive-definiteness.
Is there a sense in which the dot product is the canonical inner product on Euclidean space? Or do we just pick it because the implied norm matches our notion of distance?
linear-algebra inner-product-space
linear-algebra inner-product-space
edited 11 hours ago
Björn Friedrich
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 11 hours ago
Björn Friedrich
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 11 hours ago
Björn Friedrich
2,70661831
edited 11 hours ago
Björn Friedrich
2,70661831
edited 11 hours ago
Björn Friedrich
2,70661831
2,70661831
asked 12 hours ago
rampatowlrampatowl
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 12 hours ago
rampatowlrampatowl
1162
asked 12 hours ago
rampatowlrampatowl
1162
1162
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
rampatowl is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago
add a comment |
1
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago
1
1
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $left(e_1, frac{1}{2}e_1 + frac{sqrt{3}}{2}e_2right)$.
answered 12 hours ago
mihaildmihaild
1,03211
$endgroup$
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
add a comment |
$begingroup$
There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.
answered 12 hours ago
gandalf61gandalf61
9,293825
$endgroup$
add a comment |
$begingroup$
For an arbitrary inner product $left<right>$ on $mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = left<e_i|e_jright>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $mathbb R^n$, there is a set of real numbers $lambda_j$ and an orthonormal basis $left|xi_jright>$ such that
$$
left<a|bright> = sum_jlambda_jleft<a|xi_jright>left<xi_j|bright>
$$
Roughly speaking, the inner product resolves $a$ and $b$ into their $xi_j$ components, then weights the resulting dot product by $lambda_j$.
In general, this choice of $left|xi_jright>$ will be unique. However, for some inner products, there will be multiple possible choices of $left|xi_jright>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $left|xi_jright>$ allows the inner product to be written in that form--it is independent of the chosen basis.
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $left(e_1, frac{1}{2}e_1 + frac{sqrt{3}}{2}e_2right)$.
answered 12 hours ago
mihaildmihaild
1,03211
$endgroup$
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
add a comment |
$begingroup$
Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $left(e_1, frac{1}{2}e_1 + frac{sqrt{3}}{2}e_2right)$.
answered 12 hours ago
mihaildmihaild
1,03211
$endgroup$
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
add a comment |
$begingroup$
Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $left(e_1, frac{1}{2}e_1 + frac{sqrt{3}}{2}e_2right)$.
answered 12 hours ago
mihaildmihaild
1,03211
$endgroup$
Any inner product is dot product in some basis. For example, your inner product is standard dot product written in basis $left(e_1, frac{1}{2}e_1 + frac{sqrt{3}}{2}e_2right)$.
answered 12 hours ago
mihaildmihaild
1,03211
answered 12 hours ago
mihaildmihaild
1,03211
answered 12 hours ago
mihaildmihaild
1,03211
answered 12 hours ago
mihaildmihaild
1,03211
1,03211
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
add a comment |
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
No it's not. $e_2 = (-1,2)/sqrt{3}$ in that basis, which has euclidean norm 5/3. But $left<e_2,e_2right> = 1$.
$endgroup$
– eyeballfrog
9 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
$begingroup$
Using author's inner product we have $langle (-1, 2) / sqrt{3}, (-1, 2) / sqrt{3}rangle = 1$. And in general - if we write two vectors in this basis and take inner product as defined in question, we get their standard dot product.
$endgroup$
– mihaild
8 hours ago
add a comment |
$begingroup$
There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.
answered 12 hours ago
gandalf61gandalf61
9,293825
$endgroup$
add a comment |
$begingroup$
There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.
answered 12 hours ago
gandalf61gandalf61
9,293825
$endgroup$
add a comment |
$begingroup$
There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.
answered 12 hours ago
gandalf61gandalf61
9,293825
$endgroup$
There is nothing special about the dot product. Yes, it corresponds to the Euclidean norm if you are using an orthonormal basis. But if your basis is not orthonormal then the Euclidean norm will be represented by some other symmetric matrix.
answered 12 hours ago
gandalf61gandalf61
9,293825
answered 12 hours ago
gandalf61gandalf61
9,293825
answered 12 hours ago
gandalf61gandalf61
9,293825
answered 12 hours ago
gandalf61gandalf61
9,293825
9,293825
add a comment |
add a comment |
$begingroup$
For an arbitrary inner product $left<right>$ on $mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = left<e_i|e_jright>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $mathbb R^n$, there is a set of real numbers $lambda_j$ and an orthonormal basis $left|xi_jright>$ such that
$$
left<a|bright> = sum_jlambda_jleft<a|xi_jright>left<xi_j|bright>
$$
Roughly speaking, the inner product resolves $a$ and $b$ into their $xi_j$ components, then weights the resulting dot product by $lambda_j$.
In general, this choice of $left|xi_jright>$ will be unique. However, for some inner products, there will be multiple possible choices of $left|xi_jright>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $left|xi_jright>$ allows the inner product to be written in that form--it is independent of the chosen basis.
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
$endgroup$
add a comment |
$begingroup$
For an arbitrary inner product $left<right>$ on $mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = left<e_i|e_jright>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $mathbb R^n$, there is a set of real numbers $lambda_j$ and an orthonormal basis $left|xi_jright>$ such that
$$
left<a|bright> = sum_jlambda_jleft<a|xi_jright>left<xi_j|bright>
$$
Roughly speaking, the inner product resolves $a$ and $b$ into their $xi_j$ components, then weights the resulting dot product by $lambda_j$.
In general, this choice of $left|xi_jright>$ will be unique. However, for some inner products, there will be multiple possible choices of $left|xi_jright>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $left|xi_jright>$ allows the inner product to be written in that form--it is independent of the chosen basis.
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
$endgroup$
add a comment |
$begingroup$
For an arbitrary inner product $left<right>$ on $mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = left<e_i|e_jright>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $mathbb R^n$, there is a set of real numbers $lambda_j$ and an orthonormal basis $left|xi_jright>$ such that
$$
left<a|bright> = sum_jlambda_jleft<a|xi_jright>left<xi_j|bright>
$$
Roughly speaking, the inner product resolves $a$ and $b$ into their $xi_j$ components, then weights the resulting dot product by $lambda_j$.
In general, this choice of $left|xi_jright>$ will be unique. However, for some inner products, there will be multiple possible choices of $left|xi_jright>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $left|xi_jright>$ allows the inner product to be written in that form--it is independent of the chosen basis.
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
$endgroup$
For an arbitrary inner product $left<right>$ on $mathbb R^n$, there is a positive definite real symmetric matrix $A_{ij} = left<e_i|e_jright>$ that defines the transform. Since it is real and symmetric, it is orthogonally diagonalizable. That is, for any inner product on $mathbb R^n$, there is a set of real numbers $lambda_j$ and an orthonormal basis $left|xi_jright>$ such that
$$
left<a|bright> = sum_jlambda_jleft<a|xi_jright>left<xi_j|bright>
$$
Roughly speaking, the inner product resolves $a$ and $b$ into their $xi_j$ components, then weights the resulting dot product by $lambda_j$.
In general, this choice of $left|xi_jright>$ will be unique. However, for some inner products, there will be multiple possible choices of $left|xi_jright>$. The Euclidean norm is unique (up to a constant scaling) in that every choice of $left|xi_jright>$ allows the inner product to be written in that form--it is independent of the chosen basis.
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
answered 9 hours ago
eyeballfrogeyeballfrog
7,222633
7,222633
add a comment |
add a comment |
rampatowl is a new contributor. Be nice, and check out our Code of Conduct.
rampatowl is a new contributor. Be nice, and check out our Code of Conduct.
rampatowl is a new contributor. Be nice, and check out our Code of Conduct.
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1
$begingroup$
Not quite what you're asking - but we do know that any two inner products on a finite-dimensional vector space are equivalent, which means there are positive constants $c, C$ such that $c langle x, y rangle_1 le langle x, y rangle_2 le C langle x, y rangle_2$ for all $x,y$. So although the inner product is not unique, at least any two are within a constant scaling factor of each other. (This fact is most useful when studying a topology induced by the inner product - it means the corresponding topology doesn't depend on the choice of inner product.)
$endgroup$
– Daniel Schepler
11 hours ago