20191115_eigenvalue.Rmd

---
title: "20191115_eigenvalue"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

Checking "Eigenvectors from Eigenvalues" (arXiv:1908.03795) by Peter B. Denton, Stephen J. Parke, Terence Tao, Xining Zhang

## prepare a matrix
```{r}
n = 5
A = matrix(runif(n*n),n,n)
A = A + t(A)
A
```

## eigenvalues and eigenvectors
```{r}
f = eigen(A)
X = (f$vectors)^2 # X[j,i] = |v_{i,j}|^2
```

## eigenvalues of submatrices
```{r}
fj = list()
for(j in 1:n) fj[[j]] = eigen(A[-j,-j])
```

## reconstruct eigenvectors from eigenvalues
```{r}
Y = X
for(i in 1:n) {
  a = prod(f$values[i] - f$values[-i])
  for(j in 1:n) {
    b = prod(f$values[i] - fj[[j]]$values)
    Y[j,i] = b/a
  }
}
```

## check the result
```{r}
plot(X,Y, xlab="|v_{i,j}|^2", ylab="reconstructed from eigenvalues") ; abline(0,1, col="red")
```