Characteristic Function

Packages needed for this blog

library(ggplot2)
library(manipulate)
library(firatheme) # https://github.com/vankesteren/firatheme
library(Massign)   # https://github.com/vankesteren/Massign

The characteristic function

A while ago, I blogged that a matrix \(A\) can be seen as an operation and that the determinant of this matrix \(|A|\) says something about the volume of the transformation. This blog is about another property of matrices: the characteristic function.

The characteristic function of a square matrix \(A\) of order \(n\) is defined as follows:

\[p_A(\lambda) = | \lambda I_n - A |\]

Let’s look at this function for the following \(2\times 2\) matrix:

\[A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}\] \[ \begin{align} p_A(\lambda) &= | \lambda I_n - A | \\ &= \left| \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} - \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \right| \\ &= \left| \begin{bmatrix} \lambda - 3 & -1 \\ -2 & \lambda - 4 \end{bmatrix} \right|\\ \end{align}\]

The roots of this function - a polynomial of order \(n\) - are the eigenvalues of the matrix. We can find them using some algebra, remembering that the determinant of a \(2\times 2\) matrix is calculated as \(ad-bc\):

\[ \begin{align} p_A(\lambda) &= (\lambda - 3)(\lambda - 4) + 2\\ &= \lambda^2 - 7\lambda + 10 \\ &= (\lambda - 2)(\lambda - 5) \end{align}\]

So \(\lambda = 2\) or \(\lambda = 5\). These are the two eigenvalues of this matrix.

Visualising the characteristic function

Using the power of R, we can get a better intuition for the characteristic function by visualising it. Below the code for a function that takes a matrix and visualises this function.

charfun <- function(mat, from, to) {
  n <- ncol(mat)
  stopifnot(n == nrow(mat))
  x <- seq(from, to, length.out = 3000)
  
  ggdat <- data.frame(
    x = x,
    y = vapply(x, function(lambda) det(lambda*diag(n) - mat), 1.0)
  )
  ev <- eigen(mat)$values
  ev <- ev[ev <= to & ev > from]
  
  evdat <- data.frame(x = ev, y = rep(0, length(ev)))
  
  ggplot(ggdat, aes(x = x, y = y)) + 
    geom_hline(yintercept = 0, col = firaCols[5], lwd = 1) +
    geom_vline(xintercept = ev, col = firaCols[2], lwd = 1, lty = 2) +
    geom_line(col = firaCols[1], lwd = 1) + 
    geom_point(aes(x, y), evdat, size = 3, col = firaCols[1]) +
    labs(x = "Lambda", 
         y = "Characteristic function value", 
         title = "Characteristic function of a matrix") +
    theme_fira()
}

A %<-% "3, 1
        2, 4"

charfun(A, 0, 7) + ggtitle("Characteristic function of A")

What can we see?

  1. The characteristic function crosses the axis at 2 and 5, just as we expected
  2. The characteristic function is indeed a quadratic function, i.e., a polynomial of order \(n = 2\).

You can play around with this function in R by trying out different matrices. Try out a covariance matrix, different kinds of symmetric and assymmetric matrices!

Bonus: interactive

After running the above R chunks, you can run the following to play around with different covariance matrices of the following form:

\[A = \begin{bmatrix} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{bmatrix}\] Play around with it to see what happens to the eigenvalues of this matrix! For example, note that when a, b, c, and d are all 0 the eigenvalues are all 1. There are some nice symmetries to be explored here.

manipulate(
  {
    A %<-% " 1,
             a,  1
             b,  c,  1"
    charfun(A, 0, 2) + ylim(c(-.7,.7))
  },
  a = slider(-1, 1, initial = .5, step = 0.1),
  b = slider(-1, 1, initial = .3, step = 0.1),
  c = slider(-1, 1, initial = .2, step = 0.1)
)