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Eigenvalues of quadratic forms defined on an arbitrary inner product
Clash Royale CLAN TAG#URR8PPP
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I have problems with a generalized quadratic form defined on a non-default, multi dimensional inner product and the meaning of its eigenvalues.
Consider the following quadratic form $A(x)$
$A(x_1,x_2) = alpha |x_1|^2 - 2(1-alpha) langle x_1,x_2 rangle + alpha |x_2|^2$
where $|cdot|$ is a norm deduced from some arbitrary inner product $langle cdot, cdot rangle: H times H mapsto mathbbR$ ($|x_i|^2 = langle x_i,x_i rangle$) with a Hilbert space $H$.
In the source* it is said that there exist the two eigenvalues $1$ and $2 alpha - 1$ of $A$ and that $A$ is strictly elliptic if and only if $alpha > 1/2$.
Usually in the literature quadratic forms are written as matrix vector multiplication or standard scalar products in $mathbbR^n$, i.e. $x^T B x = x cdot (Bx) = sum_i,j=1^2 b_ij x_i x_j$
Therefore I'm looking for another inner product, we call it $[cdot,cdot]: H^2 times H^2 mapsto mathbbR$, such that one can write
$A(x) = [x, Bx] = [(x_1, x_2)^T, B(x_1,x_2)^T]$
These are my own thoughts so far:
Obviously $A$ can also be denoted as $A(x) = langle x_1, alpha x_1 rangle + 2 langle x_1, (1-alpha) x_2 rangle + langle x_2, alpha x_2 rangle$
I define $[cdot,cdot]$ as a sum of the one-dimensional inner products $<cdot,cdot>$ on $H$:
$[x,y] = [(x_1, x_2)^T , (y_1, y_2)^T] := langle x_1, y_1 rangle + langle x_2, y_2 rangle$
Then there holds
$A(x) = [x,Bx]$ with $B = beginbmatrix alpha & alpha-1 \ alpha-1 & alpha endbmatrix$. The eigenvalues of $B$ are then equal to the eigenvalues of $A$ mentioned at the beginning.
So returning to my questions:
What is the geometrical meaning of these eigenvalues? $A$ maps to a scalar and not to a vector.
Is there any generalized definition of quadratic forms on arbitrary inner products or theoreom about representation as an arbitrary inner product? The previous derivation of this kind of quadratic form is just my "invention".
Thanks!
The source is "MODELING FRACTURES AND BARRIERS AS INTERFACES FOR
FLOW IN POROUS MEDIA", published in "SIAM J. SCI. COMPUT. Society for Industrial and Applied Mathematics Vol. 26, No. 5, pp. 1667–1691", page 1676, 1st paragraph
https://epubs.siam.org/doi/pdf/10.1137/S1064827503429363
linear-algebra functional-analysis
asked Jul 30 at 21:56
mueller_seb
62
add a comment |Â
up vote
1
down vote
favorite
I have problems with a generalized quadratic form defined on a non-default, multi dimensional inner product and the meaning of its eigenvalues.
Consider the following quadratic form $A(x)$
$A(x_1,x_2) = alpha |x_1|^2 - 2(1-alpha) langle x_1,x_2 rangle + alpha |x_2|^2$
where $|cdot|$ is a norm deduced from some arbitrary inner product $langle cdot, cdot rangle: H times H mapsto mathbbR$ ($|x_i|^2 = langle x_i,x_i rangle$) with a Hilbert space $H$.
In the source* it is said that there exist the two eigenvalues $1$ and $2 alpha - 1$ of $A$ and that $A$ is strictly elliptic if and only if $alpha > 1/2$.
Usually in the literature quadratic forms are written as matrix vector multiplication or standard scalar products in $mathbbR^n$, i.e. $x^T B x = x cdot (Bx) = sum_i,j=1^2 b_ij x_i x_j$
Therefore I'm looking for another inner product, we call it $[cdot,cdot]: H^2 times H^2 mapsto mathbbR$, such that one can write
$A(x) = [x, Bx] = [(x_1, x_2)^T, B(x_1,x_2)^T]$
These are my own thoughts so far:
Obviously $A$ can also be denoted as $A(x) = langle x_1, alpha x_1 rangle + 2 langle x_1, (1-alpha) x_2 rangle + langle x_2, alpha x_2 rangle$
I define $[cdot,cdot]$ as a sum of the one-dimensional inner products $<cdot,cdot>$ on $H$:
$[x,y] = [(x_1, x_2)^T , (y_1, y_2)^T] := langle x_1, y_1 rangle + langle x_2, y_2 rangle$
Then there holds
$A(x) = [x,Bx]$ with $B = beginbmatrix alpha & alpha-1 \ alpha-1 & alpha endbmatrix$. The eigenvalues of $B$ are then equal to the eigenvalues of $A$ mentioned at the beginning.
So returning to my questions:
What is the geometrical meaning of these eigenvalues? $A$ maps to a scalar and not to a vector.
Is there any generalized definition of quadratic forms on arbitrary inner products or theoreom about representation as an arbitrary inner product? The previous derivation of this kind of quadratic form is just my "invention".
Thanks!
The source is "MODELING FRACTURES AND BARRIERS AS INTERFACES FOR
FLOW IN POROUS MEDIA", published in "SIAM J. SCI. COMPUT. Society for Industrial and Applied Mathematics Vol. 26, No. 5, pp. 1667–1691", page 1676, 1st paragraph
https://epubs.siam.org/doi/pdf/10.1137/S1064827503429363
linear-algebra functional-analysis
asked Jul 30 at 21:56
mueller_seb
62
2
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by thelangleandranglecommands, not by the "less" and "greater" inequality signs.
– zipirovich
Jul 30 at 22:12
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have problems with a generalized quadratic form defined on a non-default, multi dimensional inner product and the meaning of its eigenvalues.
Consider the following quadratic form $A(x)$
$A(x_1,x_2) = alpha |x_1|^2 - 2(1-alpha) langle x_1,x_2 rangle + alpha |x_2|^2$
where $|cdot|$ is a norm deduced from some arbitrary inner product $langle cdot, cdot rangle: H times H mapsto mathbbR$ ($|x_i|^2 = langle x_i,x_i rangle$) with a Hilbert space $H$.
In the source* it is said that there exist the two eigenvalues $1$ and $2 alpha - 1$ of $A$ and that $A$ is strictly elliptic if and only if $alpha > 1/2$.
Usually in the literature quadratic forms are written as matrix vector multiplication or standard scalar products in $mathbbR^n$, i.e. $x^T B x = x cdot (Bx) = sum_i,j=1^2 b_ij x_i x_j$
Therefore I'm looking for another inner product, we call it $[cdot,cdot]: H^2 times H^2 mapsto mathbbR$, such that one can write
$A(x) = [x, Bx] = [(x_1, x_2)^T, B(x_1,x_2)^T]$
These are my own thoughts so far:
Obviously $A$ can also be denoted as $A(x) = langle x_1, alpha x_1 rangle + 2 langle x_1, (1-alpha) x_2 rangle + langle x_2, alpha x_2 rangle$
I define $[cdot,cdot]$ as a sum of the one-dimensional inner products $<cdot,cdot>$ on $H$:
$[x,y] = [(x_1, x_2)^T , (y_1, y_2)^T] := langle x_1, y_1 rangle + langle x_2, y_2 rangle$
Then there holds
$A(x) = [x,Bx]$ with $B = beginbmatrix alpha & alpha-1 \ alpha-1 & alpha endbmatrix$. The eigenvalues of $B$ are then equal to the eigenvalues of $A$ mentioned at the beginning.
So returning to my questions:
What is the geometrical meaning of these eigenvalues? $A$ maps to a scalar and not to a vector.
Is there any generalized definition of quadratic forms on arbitrary inner products or theoreom about representation as an arbitrary inner product? The previous derivation of this kind of quadratic form is just my "invention".
Thanks!
The source is "MODELING FRACTURES AND BARRIERS AS INTERFACES FOR
FLOW IN POROUS MEDIA", published in "SIAM J. SCI. COMPUT. Society for Industrial and Applied Mathematics Vol. 26, No. 5, pp. 1667–1691", page 1676, 1st paragraph
https://epubs.siam.org/doi/pdf/10.1137/S1064827503429363
linear-algebra functional-analysis
asked Jul 30 at 21:56
mueller_seb
62
I have problems with a generalized quadratic form defined on a non-default, multi dimensional inner product and the meaning of its eigenvalues.
Consider the following quadratic form $A(x)$
$A(x_1,x_2) = alpha |x_1|^2 - 2(1-alpha) langle x_1,x_2 rangle + alpha |x_2|^2$
where $|cdot|$ is a norm deduced from some arbitrary inner product $langle cdot, cdot rangle: H times H mapsto mathbbR$ ($|x_i|^2 = langle x_i,x_i rangle$) with a Hilbert space $H$.
In the source* it is said that there exist the two eigenvalues $1$ and $2 alpha - 1$ of $A$ and that $A$ is strictly elliptic if and only if $alpha > 1/2$.
Usually in the literature quadratic forms are written as matrix vector multiplication or standard scalar products in $mathbbR^n$, i.e. $x^T B x = x cdot (Bx) = sum_i,j=1^2 b_ij x_i x_j$
Therefore I'm looking for another inner product, we call it $[cdot,cdot]: H^2 times H^2 mapsto mathbbR$, such that one can write
$A(x) = [x, Bx] = [(x_1, x_2)^T, B(x_1,x_2)^T]$
These are my own thoughts so far:
Obviously $A$ can also be denoted as $A(x) = langle x_1, alpha x_1 rangle + 2 langle x_1, (1-alpha) x_2 rangle + langle x_2, alpha x_2 rangle$
I define $[cdot,cdot]$ as a sum of the one-dimensional inner products $<cdot,cdot>$ on $H$:
$[x,y] = [(x_1, x_2)^T , (y_1, y_2)^T] := langle x_1, y_1 rangle + langle x_2, y_2 rangle$
Then there holds
$A(x) = [x,Bx]$ with $B = beginbmatrix alpha & alpha-1 \ alpha-1 & alpha endbmatrix$. The eigenvalues of $B$ are then equal to the eigenvalues of $A$ mentioned at the beginning.
So returning to my questions:
What is the geometrical meaning of these eigenvalues? $A$ maps to a scalar and not to a vector.
Is there any generalized definition of quadratic forms on arbitrary inner products or theoreom about representation as an arbitrary inner product? The previous derivation of this kind of quadratic form is just my "invention".
Thanks!
The source is "MODELING FRACTURES AND BARRIERS AS INTERFACES FOR
FLOW IN POROUS MEDIA", published in "SIAM J. SCI. COMPUT. Society for Industrial and Applied Mathematics Vol. 26, No. 5, pp. 1667–1691", page 1676, 1st paragraph
https://epubs.siam.org/doi/pdf/10.1137/S1064827503429363
linear-algebra functional-analysis
asked Jul 30 at 21:56
mueller_seb
62
edited Jul 31 at 11:26
asked Jul 30 at 21:56
mueller_seb
62
asked Jul 30 at 21:56
mueller_seb
62
asked Jul 30 at 21:56
mueller_seb
62
62
2
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by thelangleandranglecommands, not by the "less" and "greater" inequality signs.
– zipirovich
Jul 30 at 22:12
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28
add a comment |Â
2
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by thelangleandranglecommands, not by the "less" and "greater" inequality signs.
– zipirovich
Jul 30 at 22:12
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28
2
2
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by the
langle and rangle commands, not by the "less" and "greater" inequality signs.– zipirovich
Jul 30 at 22:12
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by the
langle and rangle commands, not by the "less" and "greater" inequality signs.– zipirovich
Jul 30 at 22:12
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
I think that I solved the problem:
The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $lambda_i$.
Then we get
$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = sum_i lambda_i [Sx, Sx] geq minlambda_i (|x_1|^2 + |x_2|^2)$
Thanks to all!
answered Jul 31 at 15:45
mueller_seb
62
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I think that I solved the problem:
The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $lambda_i$.
Then we get
$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = sum_i lambda_i [Sx, Sx] geq minlambda_i (|x_1|^2 + |x_2|^2)$
Thanks to all!
answered Jul 31 at 15:45
mueller_seb
62
add a comment |Â
up vote
0
down vote
I think that I solved the problem:
The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $lambda_i$.
Then we get
$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = sum_i lambda_i [Sx, Sx] geq minlambda_i (|x_1|^2 + |x_2|^2)$
Thanks to all!
answered Jul 31 at 15:45
mueller_seb
62
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I think that I solved the problem:
The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $lambda_i$.
Then we get
$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = sum_i lambda_i [Sx, Sx] geq minlambda_i (|x_1|^2 + |x_2|^2)$
Thanks to all!
answered Jul 31 at 15:45
mueller_seb
62
I think that I solved the problem:
The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $lambda_i$.
Then we get
$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = sum_i lambda_i [Sx, Sx] geq minlambda_i (|x_1|^2 + |x_2|^2)$
Thanks to all!
answered Jul 31 at 15:45
mueller_seb
62
answered Jul 31 at 15:45
mueller_seb
62
answered Jul 31 at 15:45
mueller_seb
62
answered Jul 31 at 15:45
mueller_seb
62
62
add a comment |Â
add a comment |Â
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2
Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by the
langleandranglecommands, not by the "less" and "greater" inequality signs.– zipirovich
Jul 30 at 22:12
I fixed that, thanks!
– mueller_seb
Jul 30 at 22:33
You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be.
– Will Jagy
Jul 31 at 3:05
I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices.
– mueller_seb
Jul 31 at 11:28