Non-trivial intersections of row spaces in Matlab
$begingroup$
Let $W_p$ and $W_f$ be two subspaces (past and future data in matrix form: rows as basis vectors). Let x be a vector that lies in intersection of these two subspaces. Then $∃$ two coefficient vectors $a,b$ such that
$x = a^T W_p = b^T W_f$
I want to find a, b and x, avoiding trivial intersections (i.e. vectors $a$ and $b$ such that $a^T W_p=0 =b^T W_f$)
I was thinking of proceeding in this way:
$
a^T W_p-b^T W_f = 0\
[a^T -b^T] W =0 \
$
with $W =begin{bmatrix}W_p\W_fend{bmatrix}$. Is it possible to compute $a$ and $b$ based on that? maybe with QR factorization of $W$ to find the null space?
linear-algebra matrices vector-spaces intersection-theory
$endgroup$
add a comment |
$begingroup$
Let $W_p$ and $W_f$ be two subspaces (past and future data in matrix form: rows as basis vectors). Let x be a vector that lies in intersection of these two subspaces. Then $∃$ two coefficient vectors $a,b$ such that
$x = a^T W_p = b^T W_f$
I want to find a, b and x, avoiding trivial intersections (i.e. vectors $a$ and $b$ such that $a^T W_p=0 =b^T W_f$)
I was thinking of proceeding in this way:
$
a^T W_p-b^T W_f = 0\
[a^T -b^T] W =0 \
$
with $W =begin{bmatrix}W_p\W_fend{bmatrix}$. Is it possible to compute $a$ and $b$ based on that? maybe with QR factorization of $W$ to find the null space?
linear-algebra matrices vector-spaces intersection-theory
$endgroup$
add a comment |
$begingroup$
Let $W_p$ and $W_f$ be two subspaces (past and future data in matrix form: rows as basis vectors). Let x be a vector that lies in intersection of these two subspaces. Then $∃$ two coefficient vectors $a,b$ such that
$x = a^T W_p = b^T W_f$
I want to find a, b and x, avoiding trivial intersections (i.e. vectors $a$ and $b$ such that $a^T W_p=0 =b^T W_f$)
I was thinking of proceeding in this way:
$
a^T W_p-b^T W_f = 0\
[a^T -b^T] W =0 \
$
with $W =begin{bmatrix}W_p\W_fend{bmatrix}$. Is it possible to compute $a$ and $b$ based on that? maybe with QR factorization of $W$ to find the null space?
linear-algebra matrices vector-spaces intersection-theory
$endgroup$
Let $W_p$ and $W_f$ be two subspaces (past and future data in matrix form: rows as basis vectors). Let x be a vector that lies in intersection of these two subspaces. Then $∃$ two coefficient vectors $a,b$ such that
$x = a^T W_p = b^T W_f$
I want to find a, b and x, avoiding trivial intersections (i.e. vectors $a$ and $b$ such that $a^T W_p=0 =b^T W_f$)
I was thinking of proceeding in this way:
$
a^T W_p-b^T W_f = 0\
[a^T -b^T] W =0 \
$
with $W =begin{bmatrix}W_p\W_fend{bmatrix}$. Is it possible to compute $a$ and $b$ based on that? maybe with QR factorization of $W$ to find the null space?
linear-algebra matrices vector-spaces intersection-theory
linear-algebra matrices vector-spaces intersection-theory
edited Dec 21 '18 at 9:51
Betelgeuse
asked Dec 20 '18 at 10:43
BetelgeuseBetelgeuse
1214
1214
add a comment |
add a comment |
1 Answer
1
active
oldest
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$begingroup$
Taking the transpose on both sides, your equation reads
$$
W^T pmatrix{a\ -b} = 0
$$
With this in mind, the easiest approach is probably to use the nullspace function in matlab. In particular:
W = [Wp; Wf];
N = null(W');
If all you need is a non-trivial solution, take the first column of $N$ (that is, N(:,1)
) and split it into the vectors a
and -b
.
$endgroup$
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
add a comment |
Your Answer
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
active
oldest
votes
$begingroup$
Taking the transpose on both sides, your equation reads
$$
W^T pmatrix{a\ -b} = 0
$$
With this in mind, the easiest approach is probably to use the nullspace function in matlab. In particular:
W = [Wp; Wf];
N = null(W');
If all you need is a non-trivial solution, take the first column of $N$ (that is, N(:,1)
) and split it into the vectors a
and -b
.
$endgroup$
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
add a comment |
$begingroup$
Taking the transpose on both sides, your equation reads
$$
W^T pmatrix{a\ -b} = 0
$$
With this in mind, the easiest approach is probably to use the nullspace function in matlab. In particular:
W = [Wp; Wf];
N = null(W');
If all you need is a non-trivial solution, take the first column of $N$ (that is, N(:,1)
) and split it into the vectors a
and -b
.
$endgroup$
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
add a comment |
$begingroup$
Taking the transpose on both sides, your equation reads
$$
W^T pmatrix{a\ -b} = 0
$$
With this in mind, the easiest approach is probably to use the nullspace function in matlab. In particular:
W = [Wp; Wf];
N = null(W');
If all you need is a non-trivial solution, take the first column of $N$ (that is, N(:,1)
) and split it into the vectors a
and -b
.
$endgroup$
Taking the transpose on both sides, your equation reads
$$
W^T pmatrix{a\ -b} = 0
$$
With this in mind, the easiest approach is probably to use the nullspace function in matlab. In particular:
W = [Wp; Wf];
N = null(W');
If all you need is a non-trivial solution, take the first column of $N$ (that is, N(:,1)
) and split it into the vectors a
and -b
.
answered Dec 20 '18 at 20:43
OmnomnomnomOmnomnomnom
128k791184
128k791184
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
add a comment |
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
This procedure is correct in principle, but it does not give me the correct sequence of $a,b$ and $x$ unfortunately (I have already the sequence of $x$ so to check the results).
$endgroup$
– Betelgeuse
Dec 21 '18 at 9:48
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
If you already have an $x$, then you're really solving the system $$ W_p^T a = x\ W_f^T b = x\ $$ which is to say $$ pmatrix{W_p^T&0\0&W_f^T} pmatrix{a\b} = pmatrix{x\x} $$
$endgroup$
– Omnomnomnom
Dec 21 '18 at 17:23
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
Yes, I was wondering how to proceed if you don't have an $x$
$endgroup$
– Betelgeuse
Jan 3 at 13:24
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
$begingroup$
If you don't have an $x$, then what makes a pair $a,b$ "correct"?
$endgroup$
– Omnomnomnom
Jan 3 at 13:56
add a comment |
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