Applying the chain rule on vectors and matrices
$begingroup$
I need to find $frac{dy}{dx}$ for the following
y = $||A^Tx - b||_2^2$ where $A in R^{3x3}, b in R^{3x1}, x in R^{3x1}, y in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = z^Tz$ for $z in R^{3x1}$. I'm familiar with the chain rule but I've never really used it in this way. Also, I'm not sure what $R^{3x3}$ represents and how I can use it with the chain rule.
matrices derivatives vectors real-numbers chain-rule
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
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add a comment |
$begingroup$
I need to find $frac{dy}{dx}$ for the following
y = $||A^Tx - b||_2^2$ where $A in R^{3x3}, b in R^{3x1}, x in R^{3x1}, y in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = z^Tz$ for $z in R^{3x1}$. I'm familiar with the chain rule but I've never really used it in this way. Also, I'm not sure what $R^{3x3}$ represents and how I can use it with the chain rule.
matrices derivatives vectors real-numbers chain-rule
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
$endgroup$
add a comment |
$begingroup$
I need to find $frac{dy}{dx}$ for the following
y = $||A^Tx - b||_2^2$ where $A in R^{3x3}, b in R^{3x1}, x in R^{3x1}, y in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = z^Tz$ for $z in R^{3x1}$. I'm familiar with the chain rule but I've never really used it in this way. Also, I'm not sure what $R^{3x3}$ represents and how I can use it with the chain rule.
matrices derivatives vectors real-numbers chain-rule
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
$endgroup$
I need to find $frac{dy}{dx}$ for the following
y = $||A^Tx - b||_2^2$ where $A in R^{3x3}, b in R^{3x1}, x in R^{3x1}, y in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = z^Tz$ for $z in R^{3x1}$. I'm familiar with the chain rule but I've never really used it in this way. Also, I'm not sure what $R^{3x3}$ represents and how I can use it with the chain rule.
matrices derivatives vectors real-numbers chain-rule
matrices derivatives vectors real-numbers chain-rule
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
asked Jan 14 at 20:58
Brandon MacLeodBrandon MacLeod
53
53
add a comment |
add a comment |
1 Answer
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$begingroup$
Define the vector
$$z=A^Tx-b$$
Write the function in terms of this new vector. Then find its differential and gradient.
$$eqalign{
y &= z^Tz cr
dy &= 2z^Tdz = 2z^T(A^Tdx) = (2Az)^Tdx cr
frac{partial y}{partial x} &= 2Az = 2A(A^Tx-b) crcr
}$$
The symbol ${mathbb R}^{mtimes n}$ denotes a matrix of real numbers with $m$ rows and $n$ columns.
answered Jan 15 at 18:37
greggreg
8,9951824
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Define the vector
$$z=A^Tx-b$$
Write the function in terms of this new vector. Then find its differential and gradient.
$$eqalign{
y &= z^Tz cr
dy &= 2z^Tdz = 2z^T(A^Tdx) = (2Az)^Tdx cr
frac{partial y}{partial x} &= 2Az = 2A(A^Tx-b) crcr
}$$
The symbol ${mathbb R}^{mtimes n}$ denotes a matrix of real numbers with $m$ rows and $n$ columns.
answered Jan 15 at 18:37
greggreg
8,9951824
$endgroup$
add a comment |
$begingroup$
Define the vector
$$z=A^Tx-b$$
Write the function in terms of this new vector. Then find its differential and gradient.
$$eqalign{
y &= z^Tz cr
dy &= 2z^Tdz = 2z^T(A^Tdx) = (2Az)^Tdx cr
frac{partial y}{partial x} &= 2Az = 2A(A^Tx-b) crcr
}$$
The symbol ${mathbb R}^{mtimes n}$ denotes a matrix of real numbers with $m$ rows and $n$ columns.
answered Jan 15 at 18:37
greggreg
8,9951824
$endgroup$
add a comment |
$begingroup$
Define the vector
$$z=A^Tx-b$$
Write the function in terms of this new vector. Then find its differential and gradient.
$$eqalign{
y &= z^Tz cr
dy &= 2z^Tdz = 2z^T(A^Tdx) = (2Az)^Tdx cr
frac{partial y}{partial x} &= 2Az = 2A(A^Tx-b) crcr
}$$
The symbol ${mathbb R}^{mtimes n}$ denotes a matrix of real numbers with $m$ rows and $n$ columns.
answered Jan 15 at 18:37
greggreg
8,9951824
$endgroup$
Define the vector
$$z=A^Tx-b$$
Write the function in terms of this new vector. Then find its differential and gradient.
$$eqalign{
y &= z^Tz cr
dy &= 2z^Tdz = 2z^T(A^Tdx) = (2Az)^Tdx cr
frac{partial y}{partial x} &= 2Az = 2A(A^Tx-b) crcr
}$$
The symbol ${mathbb R}^{mtimes n}$ denotes a matrix of real numbers with $m$ rows and $n$ columns.
answered Jan 15 at 18:37
greggreg
8,9951824
answered Jan 15 at 18:37
greggreg
8,9951824
answered Jan 15 at 18:37
greggreg
8,9951824
answered Jan 15 at 18:37
greggreg
8,9951824
8,9951824
add a comment |
add a comment |
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