correlation matrix

Correlation matrix

The correlation matrix of $n$ random variables $X_1,\dots ,X_n$ is the matrix $C$ which is the standardized covariance matrix. The $(i,j)$ entry is computed as:

$$\begin{array}{rl}{c}_{ij}:=\text{corr}(X_i,X_j)& =\frac{\text{cov}(X_i,X_j)}{{\sigma }_i{\sigma }_j}\end{array}$$

where ${\sigma }_i$ is the standard deviation of $X_i$, ${\mu }_i$ is the mean of $X_i$.
$-1\le c_{ij}\le 1$, where the extremes indicate perfect correlation and 0 indicates no linear correlation.

The correlation matrix is symmetric (correlation is commutative) and has ones on the diagonal (each variable is perfectly correlated with itself).

For a standardized dataset, the correlation matrix is the same as the covariance matrix.

Correlation stays the same under affine transformations, which does not hold for covariance.

$$\text{Corr}(aX+b,cY+d)=\text{Corr}(X,Y)$$

Why this does not hold for covariance:

$$\text{Cov}(aX+b,cY+d)=\mathbb{E}[(aX+\cancel{b}-\mathbb{E}[aX\cancel{+b}])(cY\cancel{+d}-\mathbb{E}[cY\cancel{+d}])]=ac\text{Cov}(X,Y)$$

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