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Is the covariance between two random variables a Hermitian matrix?
Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two. Thus the covariance matrix is a Hermitian matrix. The pseudo-covariance matrix (also called relation matrix) is defined as follows.
What do you call a random variable whose covariance is zero?
Random variables whose covariance is zero are called uncorrelated. Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated. If and are independent random variables, then their covariance is zero.
Why is the variance of a random vector called the covariance matrix?
the variance of the random vector, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector
Which is an unbiased estimate of the covariance between variable and variable?
which is an estimate of the covariance between variable and variable . The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector , a vector whose j th element is one of the random variables.
Where are the terms in the complex covariance matrix?
These terms will be in the lower right triangular part of the matrix making the complex covariance matrix Hermitian and therefore the means of each column and row vector will be a real number. Thanks for contributing an answer to Cross Validated!
How to calculate the covariance of a random variable?
For simplicity, let us assume that all random variables have zero mean. Then the covariance is defined as Cov[z, w] ≡ E[ˉzw] so we have Cov[z, w] = Re[σz, w] + iIm[σz, w] = E[→z˙→w] + iE[→z˙→w ⊥] The real (imaginary) component of σz, w is the expected value of the dot (cross) product of the associated vectors →z and →w. This is the main intuition.
How is a complex random variable different from a real random variable?
Notice the complex conjugation of the second factor in the definition. In contrast to real random variables, we also define a pseudo-covariance (also called complementary variance): The second order statistics are fully characterized by the covariance and the pseudo-covariance.