In the recent paper by Tao concerning the famous eigenvalue-eigenvector identity, I need some help in understand the (xi) point under basic consistency checks.
The actual identity (2) given in the paper is :
(Eigenvector-eigenvalue identity).
$$
\left|v_{i, j}\right|^{2} \prod_{k=1 ; k \neq i}^{n}\left(\lambda_{i}(A)-\lambda_{k}(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_{i}(A)-\lambda_{k}\left(M_{j}\right)\right) \tag{2}
$$
where $A$ is an $n \times n$ Hermitian matrix, and its $n$ real eigenvalues are $\lambda_{1}(A), \ldots, \lambda_{n}(A)$, $M_{j}$ denote the $n-1 \times n-1$ minor formed from $A$ by deleting the $j^{\text {th }}$ row and column from $A$ and $v_{i, j}$ denote the $j^{\text {th }}$ component of $v_{i}$.
(xi) Relative phase information
- As mentioned in (vii) above, the phase of any individual eigenvector $v_{i}$ is arbitrary,
- therefore the relative phase between $v_{i, k}$ and $v_{j, k}, i \neq j$, is arbitrary.
- However, the relative phases between the components of any $v_{i}$, say between $v_{i, j}$ and $v_{i, k}$ for $j \neq k$, is not arbitrary.
equation (2) can be used to extract these relative phases as follows:
consider a unitary transformation on the matrix A and its eigenvectors such that $\bf v_{i, j} \rightarrow\frac{1}{\sqrt{2}}\left(v_{i, j}+\omega v_{i, k}\right)$ and $\bf v_{i, k} \rightarrow \frac{1}{\sqrt{2}}\left(v_{i, k}-\omega^{*} v_{i,j}\right)$ with $\omega=1$ or $\sqrt{-1}$ where $\omega^{*}$ is the complex conjugate of $\omega$.
Applying (2) to the original $\mathrm{A}$ and to the two unitary transformed A's, gives us the information need to extract $\arg\left(v_{i, j} v_{i, k}^{*}\right) .$
Note $p_{M_{j}}(\lambda)$ and $p_{M_{k}}(\lambda)$ are not invariant under this particular unitary transformation,
but $p_{A}^{\prime}(\lambda)$ and other $p_{M_{l}}(\lambda), l \neq j$ or $k$, are invariant.
Further more the unitarity condition that the $\sum_{i=1}^{n} v_{i, j} v_{i, k}^{*}=0$ for $j \neq k$, can also be derived in this fashion.
I have the following question from the mentioned point:
- (How to prove) Why $p_{M_{j}}(\lambda)$ and $p_{M_{k}}(\lambda)$ are not invariant while $p_{A}^{\prime}(\lambda)$ and other $p_{M_{l}}(\lambda), l \neq j$ or $k$, are invariant under the above unitary transformation.
- How the "unitarity condition that the $\sum_{i=1}^{n} v_{i, j} v_{i, k}^{*}=0$ for $j \neq k$, can also be derived in this fashion."
If you could provide me with some understanding or a proper explanation for these two points, that would be extremely useful.