Computing the derivative of an inner product
$begingroup$
I want to differentiate (1) wrt the vector $x in mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix
begin{align}
frac{1}{2} g(x)^T w(x) g(x) &&&& (1)
end{align}
Since
begin{align}
frac{1}{2} g(x)^T w(x) g(x)= langle g(x),g(x) rangle
end{align}
Thus,
begin{align}
frac{d}{dx} langle g(x),g(x) rangle = langle g(x)',g(x) rangle + langle g(x),g(x)' rangle = 2 langle g(x)',g(x) rangle = (g(x)')^T w(x)g(x)
end{align}
but the notes I study from says the outcome is $g(x)'w(x)g(x)$
Why is that?
real-analysis linear-algebra functional-analysis derivatives inner-product-space
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
$endgroup$
add a comment |
$begingroup$
I want to differentiate (1) wrt the vector $x in mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix
begin{align}
frac{1}{2} g(x)^T w(x) g(x) &&&& (1)
end{align}
Since
begin{align}
frac{1}{2} g(x)^T w(x) g(x)= langle g(x),g(x) rangle
end{align}
Thus,
begin{align}
frac{d}{dx} langle g(x),g(x) rangle = langle g(x)',g(x) rangle + langle g(x),g(x)' rangle = 2 langle g(x)',g(x) rangle = (g(x)')^T w(x)g(x)
end{align}
but the notes I study from says the outcome is $g(x)'w(x)g(x)$
Why is that?
real-analysis linear-algebra functional-analysis derivatives inner-product-space
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
$endgroup$
$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13
add a comment |
$begingroup$
I want to differentiate (1) wrt the vector $x in mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix
begin{align}
frac{1}{2} g(x)^T w(x) g(x) &&&& (1)
end{align}
Since
begin{align}
frac{1}{2} g(x)^T w(x) g(x)= langle g(x),g(x) rangle
end{align}
Thus,
begin{align}
frac{d}{dx} langle g(x),g(x) rangle = langle g(x)',g(x) rangle + langle g(x),g(x)' rangle = 2 langle g(x)',g(x) rangle = (g(x)')^T w(x)g(x)
end{align}
but the notes I study from says the outcome is $g(x)'w(x)g(x)$
Why is that?
real-analysis linear-algebra functional-analysis derivatives inner-product-space
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
$endgroup$
I want to differentiate (1) wrt the vector $x in mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix
begin{align}
frac{1}{2} g(x)^T w(x) g(x) &&&& (1)
end{align}
Since
begin{align}
frac{1}{2} g(x)^T w(x) g(x)= langle g(x),g(x) rangle
end{align}
Thus,
begin{align}
frac{d}{dx} langle g(x),g(x) rangle = langle g(x)',g(x) rangle + langle g(x),g(x)' rangle = 2 langle g(x)',g(x) rangle = (g(x)')^T w(x)g(x)
end{align}
but the notes I study from says the outcome is $g(x)'w(x)g(x)$
Why is that?
real-analysis linear-algebra functional-analysis derivatives inner-product-space
real-analysis linear-algebra functional-analysis derivatives inner-product-space
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
edited Jan 19 at 5:35
Dreamer123
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
asked Jan 19 at 5:05
Dreamer123Dreamer123
32329
32329
$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13
add a comment |
$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13
$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13
add a comment |
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$begingroup$
They forgot to indicate the transpose?
$endgroup$
– Lord Shark the Unknown
Jan 19 at 5:05
$begingroup$
yes but not just once. Many times the transpose is dropped which made me question myself. If I am missing something
$endgroup$
– Dreamer123
Jan 19 at 5:09
$begingroup$
Does $w$ also depend on $x$? You should check how $g'$ is defined. Is it the Jacobian or its transpose?
$endgroup$
– gerw
Jan 19 at 8:13