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Before you excitedly jump to coding a new algorithm to produce eigenvectors from eigenvalues, it's worth noting that: . This only applies to hermitian matr

Before you excitedly jump to coding a new algorithm to produce eigenvectors from eigenvalues, it's worth noting that: . This only applies to hermitian matrices and is therefore of limited scope. . The identity only produces the magnitudes of the coefficients of the eigenvectors. The signs of the coefficient sare still unknown. . Even if knowing only the magnitude of the coefficients is somehow useful, this is still very expensive to compute - just like Cramer rule which is of great theoretical interest, but has very little practical use when actually computing a matrix inverse or a determinant. None of the above diminishes the importance of the result, btw.

gajomi 7y ago
Came here to pretty much write these same notes. The fact that you need to know the spectrum of all the minors really limits the usefulness for solving general eigen* problems. But I can imagine that this could make a great starting point for all kinds of approximate calculations/asymptotic analyses. Regarding the first caveat that you bring up though, whereas the problem statement says you need a Hermitian matrix I think the results should generalize to non-hermitian matrices. In particular take a look at the third proof of the theorem at the end of the article. The only assumption required here is that the eigenvalues are simple (which does not preclude them being complex/coming in complex pairs). Protip: I had to read the second step in the proof a few times before I could see what was going on. Explicitly what you do here is to (i) multiple from the right by the elementary unit vector e_j (ii) set up the matrix vector inverse using Cramer's rule (iii) notice that this matrix can be permuted to have a block diagonal element equal to the minor M_i with the other block diagonal entry equal to 1 and the corresponding column equal filled with zeros (iv) Write the determinant in the block form, which simplifies to the expression involving M_i by itself then finally (v) multiply by e_j from the left to get scalar equation.
MattSayar 7y ago
Despite the limited scope, this information is "out there" now, and could potentially be crucial for some future "useful" application. Not that I think you're trying to downplay it or anything, I'm just seeing this as another example of how the internet is still a good thing.

gajomi 7y ago

Came here to pretty much write these same notes. The fact that you need to know the spectrum of all the minors really limits the usefulness for solving general eigen* problems. But I can imagine that this could make a great starting point for all kinds of approximate calculations/asymptotic analyses. Regarding the first caveat that you bring up though, whereas the problem statement says you need a Hermitian matrix I think the results should generalize to non-hermitian matrices. In particular take a look at the third proof of the theorem at the end of the article. The only assumption required here is that the eigenvalues are simple (which does not preclude them being complex/coming in complex pairs). Protip: I had to read the second step in the proof a few times before I could see what was going on. Explicitly what you do here is to (i) multiple from the right by the elementary unit vector e_j (ii) set up the matrix vector inverse using Cramer's rule (iii) notice that this matrix can be permuted to have a block diagonal element equal to the minor M_i with the other block diagonal entry equal to 1 and the corresponding column equal filled with zeros (iv) Write the determinant in the block form, which simplifies to the expression involving M_i by itself then finally (v) multiply by e_j from the left to get scalar equation.

MattSayar 7y ago

Despite the limited scope, this information is "out there" now, and could potentially be crucial for some future "useful" application. Not that I think you're trying to downplay it or anything, I'm just seeing this as another example of how the internet is still a good thing.