# What is the mean of eigenvector of a square matrix?

Based on this quora, the first eigenvector of covariance matrix is the direction in which the data varies the most, the second eigenvector is the direction of greatest variance among those that are orthogonal (perpendicular) to the first eigenvector, the third eigenvector is the direction of greatest variance among those orthogonal to the first two, and so on.

Now I want to know whether this statement is true for any square matrix or it is only true for the covariance matrix? or what kind of condition is necessary for the statement to be true?

A covariance matrix $\Sigma$ is associated with data $X_1,\cdots,X_n$ as follows: $$\Sigma_{i,j}=\text{cov}(X_i,X_j)$$ Covariance matrices are always symmetric and positive semi-definite, and indeed all symmetric positive semi-definite matrices are covariance matrices.