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How can I find the integral of a linear map of a vector?
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Much of the literature on matrix calculus deals with derivatives.
Let $x in mathbbR^n$, and $A in mathbbR^n times n$. It is known, for example, that:
$$f(x) = Ax implies nabla_x f(x) = A tag1$$
$$f(x) = x^TAx implies nabla_x f(x) = (A + A^T)x tag2$$
How do I do the opposite? In particular, if I have $f(x) = Ax$, how can I know the antiderivative?
If $A$ were symmetric, I could perhaps find a matrix $P$ such that $A = P + P^T$, which would let me use the statement $(2)$ to claim that $int dx^T Ax = x^TPx$.
What could be said about the integral for a generally asymmetric matrix A?
linear-algebra vector-analysis matrix-calculus
asked Aug 1 at 18:34
Bee San
1314
add a comment |Â
up vote
3
down vote
favorite
Much of the literature on matrix calculus deals with derivatives.
Let $x in mathbbR^n$, and $A in mathbbR^n times n$. It is known, for example, that:
$$f(x) = Ax implies nabla_x f(x) = A tag1$$
$$f(x) = x^TAx implies nabla_x f(x) = (A + A^T)x tag2$$
How do I do the opposite? In particular, if I have $f(x) = Ax$, how can I know the antiderivative?
If $A$ were symmetric, I could perhaps find a matrix $P$ such that $A = P + P^T$, which would let me use the statement $(2)$ to claim that $int dx^T Ax = x^TPx$.
What could be said about the integral for a generally asymmetric matrix A?
linear-algebra vector-analysis matrix-calculus
asked Aug 1 at 18:34
Bee San
1314
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Much of the literature on matrix calculus deals with derivatives.
Let $x in mathbbR^n$, and $A in mathbbR^n times n$. It is known, for example, that:
$$f(x) = Ax implies nabla_x f(x) = A tag1$$
$$f(x) = x^TAx implies nabla_x f(x) = (A + A^T)x tag2$$
How do I do the opposite? In particular, if I have $f(x) = Ax$, how can I know the antiderivative?
If $A$ were symmetric, I could perhaps find a matrix $P$ such that $A = P + P^T$, which would let me use the statement $(2)$ to claim that $int dx^T Ax = x^TPx$.
What could be said about the integral for a generally asymmetric matrix A?
linear-algebra vector-analysis matrix-calculus
asked Aug 1 at 18:34
Bee San
1314
Much of the literature on matrix calculus deals with derivatives.
Let $x in mathbbR^n$, and $A in mathbbR^n times n$. It is known, for example, that:
$$f(x) = Ax implies nabla_x f(x) = A tag1$$
$$f(x) = x^TAx implies nabla_x f(x) = (A + A^T)x tag2$$
How do I do the opposite? In particular, if I have $f(x) = Ax$, how can I know the antiderivative?
If $A$ were symmetric, I could perhaps find a matrix $P$ such that $A = P + P^T$, which would let me use the statement $(2)$ to claim that $int dx^T Ax = x^TPx$.
What could be said about the integral for a generally asymmetric matrix A?
linear-algebra vector-analysis matrix-calculus
asked Aug 1 at 18:34
Bee San
1314
asked Aug 1 at 18:34
Bee San
1314
asked Aug 1 at 18:34
Bee San
1314
asked Aug 1 at 18:34
Bee San
1314
1314
add a comment |Â
add a comment |Â
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