Structure of Correlation Matrix
Jump to navigation
Jump to search
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.
$\blacksquare$
This article needs proofreading. In particular: The above does not make complete sense, but it is not clear what the parameters actually are here If you believe all issues are dealt with, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): correlation matrix
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): correlation matrix