# Updating formula for the sample covariance and correlation

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If the **covariance** matrix of our data is a diagonal matrix, such that the **covariances** are zero, then this means that the variances must be equal to the eigenvalues .

This is illustrated by figure 4, where the eigenvectors are shown in green **and** magenta, **and** where the eigenvalues clearly equal the variance components of the **covariance** matrix.

Equation (13) holds for each eigenvector-eigenvalue pair of matrix .

In the 2D case, we obtain two eigenvectors *and* two eigenvalues.

In statistics this is often refered to as ‘white data’ because its *samples* are drawn from a standard normal distribution *and* therefore correspond to white (uncorrelated) noise: However, although equation (12) holds when the data is scaled in the x *and* y direction, the question rises if it also holds when a rotation is applied.

However, if the **covariance** matrix is not diagonal, such that the **covariances** are not zero, then the situation is a little more complicated.

The eigenvalues still represent the variance magnitude in the direction of the largest spread of the data, *and* the variance components of the *covariance* matrix still represent the variance magnitude in the direction of the x-axis *and* y-axis.

Each of the examples in figure 3 can simply be considered to be a linearly transformed instance of figure 6:where *and* are the scaling factors in the x direction *and* the y direction respectively.

In the following paragraphs, we will discuss the relation between the *covariance* matrix , *and* the linear transformation matrix .

So, if we would like to represent the **covariance** matrix with a vector **and** its magnitude, we should simply try to find the vector that points into the direction of the largest spread of the data, **and** whose magnitude equals the spread (variance) in this direction.

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