Structure of Correlation Matrix

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Definition

Let $\sequence a_n$ and $\sequence b_n$ be sequences of $n$ observations.

Let $\mathbf C$ be the correlation matrix with respect to $\sequence a_n$ and $\sequence b_n$.


Then $\mathbf C$ is symmetric with entries on the main diagonal all equal to $1$.


Proof

By definition of correlation matrix:

$\sqbrk c_{i j} = r_{i j}$

where $r_{i j}$ is the sample correlation coefficient between $a_i$ and $b_j$.

Hence by definition of sample correlation coefficient:

$\sqbrk c_{i j} = \dfrac {s_{a_i b_j} } {\sqrt {s_{a_i a_i} s_{b_j b_j} } }$

We see that:

\(\ds \sqbrk c_{i j}\) \(=\) \(\ds \dfrac {s_{a_i b_j} } {\sqrt {s_{a_i a_i} s_{b_j b_j} } }\)
\(\ds \) \(=\) \(\ds \dfrac {s_{b_j a_i} } {\sqrt {s_{b_j b_j} s_{a_i a_i} } }\)
\(\ds \) \(=\) \(\ds \sqbrk c_{j i}\)

demonstrating the symmetric nature of $\mathbf C$.

$\Box$


Then we note that:

\(\ds \sqbrk c_{i i}\) \(=\) \(\ds \dfrac {s_{a_i b_i} } {\sqrt {s_{a_i a_i} s_{b_i b_i} } }\)
\(\ds \) \(=\) \(\ds 1\)

Hence the result.

$\blacksquare$


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