what does eigenvalues expres in the covariance matrix?

is there a relationship between a covariance matrix and eigenvalues? like an example

Let us consider a 321 × 261 image dimention 321 × 261 = 83781. We have only 32 observations and 83781 unknowns then we have a matrix of (32 row X 83781 column)

then we will calculate the covariance matrix (32 X 32) so we get 32 eigenvalues the question is: does these eigenvalues express the 32 images? or there is no any relationship between eigenvalues and images

thanks for you,

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Long story short: The eigenvalues of the covariance matrix encode the variability of the data in an orthogonal basis that captures as much of the data’s variability as possible in the first few basis functions (aka the principle component basis).

For example, this code creates an ellipse, whos major axis is the x-axis, and whos minor axis is the y-axis.

t = linspace(0,2*pi,256);
data = [cos(t);0.2*sin(t)];
plot(data(1,:),data(2,:),'.')
axis([-1,1,-1,1])

Now, compute the variance of the data’s coordinates

% tranpose to get variance down each column
% computes variance of each coordinate of the data
v = var(data')

Finally, observe the eigenvalues of the covariance matrix are equal to the variance of the data’s coordinates

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what does eigenvalues expres in the covariance matrix?

what does eigenvalues expres in the covariance matrix?