Why is a correlation matrix symmetric?

I'm sorry for being so weak in math. (I'm a student) For eg. this is a correlation matrix.

Q1        Q2        Q3
Q1  1.000000  0.707568  0.014746
Q2  0.707568  1.000000 -0.039130
Q3  0.014746 -0.039130  1.000000

Why is it symmetric? Why is Q1:Q2, the same as Q2:Q1? Shouldn't they be inverses of each other? How do I read this and understand the relation?

• The correlation between random variables $X$ and $Y$ is $\mathbb E [(X-\mu_x)(Y-\mu_y)]/\sigma_x \sigma_y$. You should be able to answer your question using the properties of expectations of products. Welcome to the site! – Emre May 7 '18 at 17:43
• I'm using the Pearson correlation – LazyAwkwardVaish Ok May 7 '18 at 17:44

The correlation matrix is a measure of linearity. It does not express how two variables are dependent on each other. If the relationship is approximately linear, the absolute value of correlation will be closer to 1. If there is no linear relationship, the value is zero.

Consider two sets of variables (x1,y1) and (x2,y2).

y1 = 2 * x1

y2 = 1000 * x2

In both these cases, the correlation is 1.

The exact relationship between x1 and y1 cannot be understood by looking only at the correlation matrix.

• Thanks for mentioning that with example. Suppose the constant/coefficient is lesser than 1. Like y1=0.5*x1. Would the correlation become negative now? – LazyAwkwardVaish Ok May 8 '18 at 11:03
• No. If you plot y1=0.5*x1, you will get a line in x-y plane. So, in this case also correlation is 1. – Saptarshi Roy May 8 '18 at 11:09
• So what does the negative correlation between two variables indicate? – LazyAwkwardVaish Ok May 8 '18 at 11:11
• Negative correlation means if x1 increases, y1 decreases. In all the previous examples, y1 increases with increasing x1. So, in all those cases correlation were 1. An example dataset (x1,y1) would have correlation -1 if the points lie on the line y1=-2*x1. – Saptarshi Roy May 8 '18 at 11:16
• "It does not express how two variables are dependent on each other" , Is there anything that is related to if/how two variables are dependent on each other? or should I post this as a question? – jimjim Oct 30 '19 at 2:38

Intuitively, the correlation matrix is symmetric because every variable pair has to have the same relationship (correlation) whether their correlation is in the upper right or lower left triangle. It doesn’t make sense to say the correlation between variables $X_1$ and $X_2$ is $\rho$, but the correlation between $X_2$ and $X_1$ is $\rho’\neq \rho$ if calculating a Pearson correlation (so correlation is symmetric).

Mathematically, correlation between two variables, $X$ and $Y$, is commutative: $Corr(X,Y)=Corr(Y,X)$.

In OP’s case, the correlation between Q1 and Q2 is the same calculation and therefore the same result as the correlation between Q2 and Q1. Therefore the correlation matrix will be symmetric.

There are more mathematical reasons and proofs why a correlation matrix of real valued variables has to be symmetric and positive semi-definite, but I’ve excluded them from this answer.

• So if the correlation Corr(A, B) and Corr(B,A) are the same i.e 0.707658. Which is the one that is bigger? How do I know the relative magnitudes of the variables looking that the matrix? – LazyAwkwardVaish Ok May 8 '18 at 6:23
• corr(A,B)=corr(B,A) since (pearson) correlation is defined as corr(X,Y)=cov(X,Y)/(std(X)*std(Y)), and cov(X,Y) (the covariance) is commutative, so cov(X,Y)=cov(Y,X). So neither is "bigger" – PyRsquared May 8 '18 at 8:00

A correlation matrix is symmetric because it represents correlations among variables and correlation is a symmetric relation.

What is a correlation matrix? A bit more formally, for a set of $$n$$ random variables $$X_{1},\ldots ,X_{n}$$ the correlation matrix contains at place $$(i,j)$$ the value of the correlation between $$X_{i}$$ and $$X_{j}$$.

Denote by $$corr(X_{i}$$, $$X_{j})$$ the correlation between variable $$X_{i}$$ and $$X_{j}$$. From the fact that correlation is a symmetric relation we have that for every $$i,j$$ $$corr(X_{i}, X_{j}) = corr(X_{j}, X_{i})$$

that is the $$(i,j)$$ entry of the correlation matrix is equal to the $$(j,i)$$ entry and this is precisely what makes a matrix symmetric.