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Jan
25
comment What's the interpretation of a unitary matrix?
... Since $\mathbf{v}_j$ has been mapped to $\mathbf{e}_j$ by $U^\dagger$, we need to stretch $\mathbf{e}_j$ by $\lambda_j$ since that's the original action of $A$. This is implemented by the diagonal matrix $D$. Now we rotate back to get the original matrix $A$. This sequence of actions completely characterizes any diagonalizable matrix, with the only difference being that normal matrices have orthogonal eigenvectors while other diagonalizable matrices may not necessarily have orthogonal eigenvectors.
Jan
25
comment What's the interpretation of a unitary matrix?
... Note that the action of $A$ on $\mathbf{v}_j$ is to just stretch it by the corresponding eigenvalue $\lambda_j$. Since $A$ is normal, we know that the eigenvectors can be chosen to be orthonormal. Now, as usual, we form the matrix $U$ where the $j$th column is the vector $\mathbf{v}_j$. The action of $U$ is to take the $j$th standard basis vector $\mathbf{e}_j$ to the eigenvector $\mathbf{v}_j$, i.e. it is an orthogonal transformation which rotates the standard basis to the eigenbasis. If we apply the inverse $U^\dagger$ first, we rotate the eigenbasis to the standard basis.
Jan
25
comment What's the interpretation of a unitary matrix?
@Tobia Yes, you are absolutely correct that we stretch by the magnitude of the eigenvalues, in the direction of the corresponding eigenvectors. As you correctly pointed out, all vectors in the same direction of an eigenvector (i.e. scalar multiples of an eigenvector) are also eigenvectors themselves. You are correct that we need a unitary $U$ and a diagonal matrix $D$ to characterize a matrix (more precisely, a normal matrix). Suppose we have a normal matrix $A$, so that $A$ is unitarily diagonalizable. Let $\mathbf{v}_j$ denote the $j$th eigenvector of $A$.
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16
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Dec
24
accepted Elliptic functions as inverses of Elliptic integrals
Dec
23
comment Elliptic functions as inverses of Elliptic integrals
Thank you for the answer. The counter-example provided seems to be rather trivial, which is perhaps due to my poor definitions. What if we insist that the integrand must be of the form $\frac{A(x)+B(x)\sqrt{P(x)}}{C(x)+D(x)\sqrt{P(x)}}$, with $B$ or $D$ non-zero? I know of the decomposition in terms of the three elliptic integrals, but I am not interested in evaluating the elliptic integrals. I am simply curious whether the modern definition of elliptic functions as doubly periodic meromorphic functions can be realized as the inverses of elliptic integrals, as they were considered classically.
Dec
23
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Dec
15
comment Elliptic functions as inverses of Elliptic integrals
@Martín-BlasPérezPinilla I had actually seen this reference already, and it treats the elliptic integrals of the first and second kinds along with the Jacobi and Weirstrass elliptic functions. The general claim doesn't seem to be there.
Dec
15
asked Elliptic functions as inverses of Elliptic integrals
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13
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6
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22
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6
answered Finite dimensional vector space.