# Is Covariance Matrix analogous to Jacobian Matrix?

In probability theory covariance matrix denote how each variable relates to other in a pairwise manner. So 1 would mean they are identical and 0 would mean they are independent and are not related. Is this concept similar to Jacobian matrix where such relation between multiple variables are denoted by partial derivative?

• Because covariance matrices represent quadratic forms (which is why they are always symmetric and positive semi-definite) and Jacobians represent arbitrary linear transformations, about the only thing they have in common is that they are square matrices. Your question sounds like you are asking whether apples and squid are similar. They are in the sense that they are derived from living things, but there's little to recommend the comparison. – whuber Mar 5 at 16:21
• allright makes sense! +1 for creative analogy – GENIVI-LEARNER Mar 5 at 16:28
• In general, whuber is right. But there are actually some similarities. And in some cases, the covariance matrix is the jacobian matrix -- e.g. when a variable transformation is performed; to be more precise, it would need to be a linear transformation, otherwise it's only an approximation. In general it helps to think of the Jacobian simply as the derivative in spaces with dimension larger one. But I'd also be careful to associate covariance with "how a variable relates to another"; it's just the expectation value of the product. – cherub Mar 6 at 15:19
• @cherub is there any concrete example you would like to share where covariance matrix is the jacabian matrix. My initial motivation of seeking similarities between the two were due to taking point-wise partial derivative in Jacobian matrix where each element of the matrix quantify how one varies when other being kept constant. or relative increment. Can this notion be related to an equivalent element of covariance matrix where this relative increment is quantified as expectation value of the product of the corresponding variables? – GENIVI-LEARNER Mar 6 at 16:17
• @cherub much of what you write is confusing. In particular, any covariance matrix can be considered a linear transformation--that's standard, but in a statistical application it is important to describe what it might be transforming. A Jacobian cannot possibly be a covariance matrix unless it is square, symmetric, and positive-semidefinite, which in more than one dimension would be a rare circumstance, and even then it's unclear what it would be the covariance of. – whuber Mar 7 at 13:57