Before we discuss the PCA theorem, we need to remind ourselves about the definition of the covariance and the correlation function. These are quantities
Suppose we have defined two vectors ˆx and ˆy with n elements each. The covariance matrix C is defined as C[x,y]=[cov[x,x]cov[x,y]cov[y,x]cov[y,y]], where for example cov[x,y]=1nn−1∑i=0(xi−¯x)(yi−¯y). With this definition and recalling that the variance is defined as var[x]=1nn−1∑i=0(xi−¯x)2, we can rewrite the covariance matrix as C[x,y]=[var[x]cov[x,y]cov[x,y]var[y]].