Rewriting $left[begin{array}{c} I_notimes e_1\ vdots \ I_n otimes e_T end{array}right]$
Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?
$left[begin{array}{c}
I_notimes e_1\
vdots \
I_n otimes e_T end{array}right]$
$e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.
I need to rewrite this matrix to make it faster to work with when programming.
linear-algebra
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
add a comment |
Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?
$left[begin{array}{c}
I_notimes e_1\
vdots \
I_n otimes e_T end{array}right]$
$e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.
I need to rewrite this matrix to make it faster to work with when programming.
linear-algebra
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
add a comment |
Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?
$left[begin{array}{c}
I_notimes e_1\
vdots \
I_n otimes e_T end{array}right]$
$e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.
I need to rewrite this matrix to make it faster to work with when programming.
linear-algebra
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
Is there a way to rewrite the matrix below using operations like $vec$, $otimes$, etc?
$left[begin{array}{c}
I_notimes e_1\
vdots \
I_n otimes e_T end{array}right]$
$e_i$ is the $i$-th column of the matrix $I_T$, $otimes$ is the Kronecker product.
I need to rewrite this matrix to make it faster to work with when programming.
linear-algebra
linear-algebra
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
asked Nov 13 '18 at 15:52
An old man in the sea.
1,62211032
1,62211032
add a comment |
add a comment |
1 Answer
1
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Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
= begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$
Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
begin{bmatrix}
text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
end{bmatrix}.$$
This avoids having to form the full Kronecker product.
That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.
answered Nov 19 '18 at 3:35
OtZman
814
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
= begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$
Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
begin{bmatrix}
text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
end{bmatrix}.$$
This avoids having to form the full Kronecker product.
That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.
answered Nov 19 '18 at 3:35
OtZman
814
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
add a comment |
Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
= begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$
Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
begin{bmatrix}
text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
end{bmatrix}.$$
This avoids having to form the full Kronecker product.
That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.
answered Nov 19 '18 at 3:35
OtZman
814
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
add a comment |
Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
= begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$
Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
begin{bmatrix}
text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
end{bmatrix}.$$
This avoids having to form the full Kronecker product.
That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.
answered Nov 19 '18 at 3:35
OtZman
814
Is your goal to compute matrix-vector products involving this matrix? If this is the case, you could restructure the computation as follows. We have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x
= begin{bmatrix}(I_n otimes e_1)x newline (I_n otimes e_2)x newline vdots newline (I_n otimes e_T)x end{bmatrix}.$$
Using e.g. equation (2) from the Van Loan paper "The ubiquitous Kronecker product" from 2000, we can write $(I_n otimes e_i) x = text{vec}(e_i x^top)$ for each $i$. We therefore have
$$begin{bmatrix}I_n otimes e_1 newline I_n otimes e_2 newline vdots newline I_n otimes e_Tend{bmatrix} x =
begin{bmatrix}
text{vec}(e_1 x^top) newline text{vec}(e_2 x^top) newline vdots newline text{vec}(e_T x^top)
end{bmatrix}.$$
This avoids having to form the full Kronecker product.
That Van Loan paper I cited above contains lots of useful information about Kronecker products if you deal with them a lot.
answered Nov 19 '18 at 3:35
OtZman
814
answered Nov 19 '18 at 3:35
OtZman
814
answered Nov 19 '18 at 3:35
OtZman
814
answered Nov 19 '18 at 3:35
OtZman
814
814
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
add a comment |
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
Thanks OtZman ;)
– An old man in the sea.
Dec 21 '18 at 8:41
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
If you're insterested ;) math.stackexchange.com/questions/3048304/…
– An old man in the sea.
Dec 21 '18 at 8:45
add a comment |
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