A covariance matrix is a P*P matrix with elements from a vector of random variables. The element in the position i, j is the given covariance between the ith and jth elements. The covariance matrix can also be referred to as the variance-covariance matrix or the dispersion matrix. The concept of covariance matrix enhances the general idea of variance in multiple dimensions.
The dispersion matrix has the variance of the different elements as the main diagonal. All the elements off the main diagonal are the respective covariance between the elements. The covariance between the elements (i,j) is similar to (j,i) , this property is referred to us symmetry. Due to its symmetric property each covariance is displayed twice within the matrix.
In data analytics, the case of multivariate data sets is quite normal, in order to check for variation within the variables, the applications of mathematical formulas that employ the covariance concept come in handy. One of these examples is the principal component analysis.
In investment banking the covariance matrix is one of the greatest tools. It is used in portfolio diversification and optimization. The matrix reduces chances of risks within the stock market and investment banking, and also gives prediction for the highest return within the investments. For instance a portfolio with a covariance matrix, whose variance is 0, has the highest diversification hence lower risk.