I want to measure the scale of a 2d matrix that contains the coordinates (x,y) of a set of point. I know that the standard deviation of the x coordinates of these points is the scale of the x coordinates, and the same for the standard deviation of the y coordinates. I looked for a way to calculate the standard deviation of the whole matrix but apparently the standard deviation is defined for 1d vectors only. But I noticed that some people use the square of the norm of the covariance matrix as a scale. Does that mean that the covariance matrix is the equivalent of the standard deviation, but the latter is defined for 1d vectors while the former is for 2d matrices? Is there another way to calculate the scale of a set of points?
Solved – Is the covariance matrix the equivalent of standard deviation for a 2d matrix
covariance-matrixstandard deviation
Best Answer
Short answer: The covariance matrix is the multidimensional analog of 1-d variance (which is itself sd^2).
Some authors have even referred to the covariance matrix as the variance-covariance matrix, or even simply the variance where the dimensions are implied from context.
If you are looking for scale specifically, you could get the square roots of the eigenvalues of the covariance matrix which will be the standard deviations along the principal components. For this interpretation see https://math.stackexchange.com/questions/23596/why-is-the-eigenvector-of-a-covariance-matrix-equal-to-a-principal-component. The "volume" of your covariance matrix can be found by the square root of the product of these eigenvalues, which is also equal to the square root of the determinant of the matrix.
Alternatively, you might consider the Cholesky decomposition as a method to get something similar to a multivariate standard deviation. This concept is often seen when generating random variates from a multivariate normal distribution and the resulting lower triangular matrix is used essentially in place of the univariate standard deviation. See here for more details: Can I use the Cholesky-method for generating correlated random variables with given mean?