行列理論と金融工学

最近リスク管理の試験の準備で共分散行列を学んでいます。
A statement I have read but do not understand is "all covariance matrices should be positive definite". A positive definite matrix M is one for which for any vector w,
w^TMw>0 holds.
In addition (although I cannot prove this) all eigenvalues are positive.
Qualitatively, I am not sure why this matters for covariance matrices, which express the relationships between variables. Each element of a covariance matrix V(i,j) is the covariance between the ith variable and the jth variable. This leads one to ask the definition of covariance. It is:
E*1
In other words, it is the expected product of the distances from the mean of the two variables. In financial analysis, this might be used to indicate the relationships between the underlying assets of a portfolio. The diagonal entries of this matrix are the variance of the ith variable.
However, I still do not see what property of the matrix causes it to be positive definite.
An example of a non-positive-definite matrix is [[1/4,1/4],[1/4,1/4]], as if w=[1,-1], the w^TVw product is 0. I suppose such a matrix could arise as a covariance matrix. However, I am not sure why this would pose a problem qualitatively. In terms of applications, a matrix must be positive definite in order for a Cholesky decomposition to be feasible, it seems.
An example of a non-positive-definite covariance case would be where x2=2*x1, as then cov=2*var(x1), and the matrix is of the form [[1,2],[1,2]]. Using w=[1,-1] results in w^TVw=-2. Note that, for any real vector w, w^Tw>=0, because negative terms will be multiplied together in the dot product. Qualitatively, it seems to be that a positive definite matrix, when multiplied by an eigenvector, will not reverse its direction. A non-positive definite matrix might. Considering the matrix above, x1 varies more in combination with x2 than it does in combination with itself. This causes the covariance matrix to emphasize the second coordinate of w in setting the first coordinate of the transformed vector. Why this is undesirable for the Cholesky decomposition is another matter, but it does seem that, from a financial perspective, x2 is a redundant variable.

A related issue is the correlation coefficient: p = s12/(s1)(s2), where s12 is the covariance of variables 1 and 2, while s1 and s2 are their standard deviations. I have yet to prove mathematically that a value of 1 for this means that the variables move in synch, or even why the range of this seems to be [-1,1].

*1:X[i]-mu[i])(X[j]-mu[j]