Eigenvector relations between matrices whose Gramian Matrices are the same
up vote
0
down vote
favorite
I have two symmetric real matrices I am interested in, $A_{ntimes n}$ and $B_{mtimes m}$. If I do the following operations:
$$EAE^{T}=FBF^{T}$$
where $E$ and $F$ are of dimensions $ptimes n$ and $ptimes m$ respectively.
Supposed that the $A$ and $B$ can be eigen-decomposed as $A=Q_ALambda_A Q_A^{T}$, $B=Q_BLambda_B Q_B^{T}$, and thus the previous equation can be written as
$$EQ_ALambda_A Q_A^{T}E^{T}=FQ_BLambda_B Q_B^{T}F^{T}$$
where $Q$ and $Lambda$ are matrices containing eigenvectors and eigenvalues, respectively
or in the form as
$$(EQ_ALambda_A^{1/2}) (EQ_ALambda_A^{1/2} )^T=(FQ_BLambda_B^{1/2}) (FQ_BLambda_B^{1/2} )^T.$$
If I further let $K=EQ_ALambda_A^{1/2}$ and $L=FQ_BLambda_B^{1/2}$, then I obtain
$$KK^T=LL^T.$$
Here come my questions:
- What are the relations between $Q_A$ and $Q_B$?
- To simplify the first question, my strategy is to obtain $KK^{T}=LL^{T}$ as mentioned, and find the relations between $K$ and $L$ first. That is, what are the relations between matrices with the same Gramian matrix?
I found the answer in wiki, which says, "...in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix." But I still can not understand what it exactly means.
Thus, can anyone help me with these two questions or give me any references to read?
linear-algebra matrix-decomposition matrix-analysis
asked Nov 14 at 19:19
Charlie
12
add a comment |
up vote
0
down vote
favorite
I have two symmetric real matrices I am interested in, $A_{ntimes n}$ and $B_{mtimes m}$. If I do the following operations:
$$EAE^{T}=FBF^{T}$$
where $E$ and $F$ are of dimensions $ptimes n$ and $ptimes m$ respectively.
Supposed that the $A$ and $B$ can be eigen-decomposed as $A=Q_ALambda_A Q_A^{T}$, $B=Q_BLambda_B Q_B^{T}$, and thus the previous equation can be written as
$$EQ_ALambda_A Q_A^{T}E^{T}=FQ_BLambda_B Q_B^{T}F^{T}$$
where $Q$ and $Lambda$ are matrices containing eigenvectors and eigenvalues, respectively
or in the form as
$$(EQ_ALambda_A^{1/2}) (EQ_ALambda_A^{1/2} )^T=(FQ_BLambda_B^{1/2}) (FQ_BLambda_B^{1/2} )^T.$$
If I further let $K=EQ_ALambda_A^{1/2}$ and $L=FQ_BLambda_B^{1/2}$, then I obtain
$$KK^T=LL^T.$$
Here come my questions:
- What are the relations between $Q_A$ and $Q_B$?
- To simplify the first question, my strategy is to obtain $KK^{T}=LL^{T}$ as mentioned, and find the relations between $K$ and $L$ first. That is, what are the relations between matrices with the same Gramian matrix?
I found the answer in wiki, which says, "...in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix." But I still can not understand what it exactly means.
Thus, can anyone help me with these two questions or give me any references to read?
linear-algebra matrix-decomposition matrix-analysis
asked Nov 14 at 19:19
Charlie
12
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have two symmetric real matrices I am interested in, $A_{ntimes n}$ and $B_{mtimes m}$. If I do the following operations:
$$EAE^{T}=FBF^{T}$$
where $E$ and $F$ are of dimensions $ptimes n$ and $ptimes m$ respectively.
Supposed that the $A$ and $B$ can be eigen-decomposed as $A=Q_ALambda_A Q_A^{T}$, $B=Q_BLambda_B Q_B^{T}$, and thus the previous equation can be written as
$$EQ_ALambda_A Q_A^{T}E^{T}=FQ_BLambda_B Q_B^{T}F^{T}$$
where $Q$ and $Lambda$ are matrices containing eigenvectors and eigenvalues, respectively
or in the form as
$$(EQ_ALambda_A^{1/2}) (EQ_ALambda_A^{1/2} )^T=(FQ_BLambda_B^{1/2}) (FQ_BLambda_B^{1/2} )^T.$$
If I further let $K=EQ_ALambda_A^{1/2}$ and $L=FQ_BLambda_B^{1/2}$, then I obtain
$$KK^T=LL^T.$$
Here come my questions:
- What are the relations between $Q_A$ and $Q_B$?
- To simplify the first question, my strategy is to obtain $KK^{T}=LL^{T}$ as mentioned, and find the relations between $K$ and $L$ first. That is, what are the relations between matrices with the same Gramian matrix?
I found the answer in wiki, which says, "...in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix." But I still can not understand what it exactly means.
Thus, can anyone help me with these two questions or give me any references to read?
linear-algebra matrix-decomposition matrix-analysis
asked Nov 14 at 19:19
Charlie
12
I have two symmetric real matrices I am interested in, $A_{ntimes n}$ and $B_{mtimes m}$. If I do the following operations:
$$EAE^{T}=FBF^{T}$$
where $E$ and $F$ are of dimensions $ptimes n$ and $ptimes m$ respectively.
Supposed that the $A$ and $B$ can be eigen-decomposed as $A=Q_ALambda_A Q_A^{T}$, $B=Q_BLambda_B Q_B^{T}$, and thus the previous equation can be written as
$$EQ_ALambda_A Q_A^{T}E^{T}=FQ_BLambda_B Q_B^{T}F^{T}$$
where $Q$ and $Lambda$ are matrices containing eigenvectors and eigenvalues, respectively
or in the form as
$$(EQ_ALambda_A^{1/2}) (EQ_ALambda_A^{1/2} )^T=(FQ_BLambda_B^{1/2}) (FQ_BLambda_B^{1/2} )^T.$$
If I further let $K=EQ_ALambda_A^{1/2}$ and $L=FQ_BLambda_B^{1/2}$, then I obtain
$$KK^T=LL^T.$$
Here come my questions:
- What are the relations between $Q_A$ and $Q_B$?
- To simplify the first question, my strategy is to obtain $KK^{T}=LL^{T}$ as mentioned, and find the relations between $K$ and $L$ first. That is, what are the relations between matrices with the same Gramian matrix?
I found the answer in wiki, which says, "...in finite-dimensions it determines the vectors up to isomorphism, i.e. any two sets of vectors with the same Gramian matrix must be related by a single unitary matrix." But I still can not understand what it exactly means.
Thus, can anyone help me with these two questions or give me any references to read?
linear-algebra matrix-decomposition matrix-analysis
linear-algebra matrix-decomposition matrix-analysis
asked Nov 14 at 19:19
Charlie
12
asked Nov 14 at 19:19
Charlie
12
edited Nov 15 at 14:14
asked Nov 14 at 19:19
Charlie
12
asked Nov 14 at 19:19
Charlie
12
asked Nov 14 at 19:19
Charlie
12
12
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54
add a comment |
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998705%2feigenvector-relations-between-matrices-whose-gramian-matrices-are-the-same%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
The statement on the wiki is saying that if $A A^T=B B^T$ then there exists an orthogonal matrix $Q$ with $A=BQ$.
– Ian
Nov 14 at 19:54