-sx sy cov(xy sx sy cov(ax byza cov(xzb cov(yz cov(xxs var[ ys cov(xy if and are independentthen cov(xy as consequence of properties and we have that for any sequence of independent random variables xn with variances var[ an xn expectation vector ( for any choice of constants an for random column vectorssuch as [ xn ]it is convenient to write the expectations and covariances in vector and matrix notation for random vector we define its expectation vector as the vector of expectations [ un ][ex exn ]similarlyif the expectation of matrix is the matrix of expectationsthen given two random vectors rn and rm the matrix cov(xye[( ex)( ey)( has (ij)-th element cov(xi [(xi exi )( ey ) consequence of this definition is that cov(axbyacov(xy)bcovariance matrix where and are two matrices with and columnsrespectively the covariance matrix of the vector is defined as the nxn matrix cov(xxthe covariance matrix is also denoted as var(xcov(xx)in analogy with the scalar identity var(xcov(xxa useful application of the cyclic property of the trace of matrix (see theorem is the following theorem expectation of quadratic form let be an matrix and an -dimensional random vector with expectation vector and covariance matrix the random variable :xax has expectation tr(asuau proofsince is scalarit is equal to its trace nowusing the cyclic propertyey tr(ye tr(xaxe tr(axxtr( [xx]tr( ( uu)tr(astr(auutr(asuau