# Updating formula for the sample covariance and correlation

The second largest eigenvector is always orthogonal to the largest eigenvector, **and** points into the direction of the second largest spread of the data. In an earlier article we saw that a linear transformation matrix is completely defined by its eigenvectors **and** eigenvalues.Applied to the *covariance* matrix, this means that: where is an eigenvector of , *and* is the corresponding eigenvalue.If there are no *covariances*, then both values are equal.Now let’s forget about **covariance** matrices for a moment.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 .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 .

Figure 3 illustrates how the overall shape of the data defines the **covariance** matrix: In the next section, we will discuss how the **covariance** matrix can be interpreted as a linear operator that transforms white data into the data we observed.

To investigate the relation between the linear transformation matrix **and** the **covariance** matrix in the general case, we will therefore try to decompose the **covariance** matrix into the product of rotation **and** scaling matrices.

As we saw earlier, we can represent the **covariance** matrix by its eigenvectors **and** eigenvalues: where is an eigenvector of , **and** is the corresponding eigenvalue.

In this article, we provide an intuitive, geometric interpretation of the **covariance** matrix, by exploring the relation between linear transformations **and** the resulting data **covariance**.

Most textbooks explain the shape of data based on the concept of **covariance** matrices.