For an inverted correlation matrix $C^{-1}$, I read that its diagonal elements  are related to the multiple correlation between measure i as a criterion predicted from all other measures in the set, as :  $$ R_{i,12...p} = \sqrt{1 - \frac{1}{C^{-1}_{ii}}} $$  Why is that?  There is a related question . But that is about the off-diagonal elements.
122929
I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in , but without a proof).  Why is this the case?
38831
Find the correlation coefficient $\rho_g$ of $(G(x), H(y))$ where $G(x),H(y)$ are marginal  C.D.Fs of $X,Y$ and $X,Y$ follow  bivariate normal distribution $\text{B.N.}(\mu_1,\mu_2,\sigma_1,\sigma_2,\rho)$. I am finding it difficult to calculate $E(G(x)H(y))$, otherwise I can find all the other terms. Please help! The answer given is $\frac{6}{\pi} \arcsin{\frac{\rho}{2}}$
25624
I have several hundred measurements. Now, I am considering utilizing some kind of software to correlate every measure with every measure. This means that there are thousands of correlations. Among these there should (statistically) be a high correlation, even if the data is completely random (each measure has only about 100 datapoints).  When I find a correlation, how do I include the information about how hard I looked for a correlation, into it?   I am not at a high level in statistics, so please bear with me.
21230
Framework: From RMT, eigenvalues have a semicircle distribution for symmetric matrices each with i.i.d normally distributed entries as the size of the matrix grows. The restrictions on i.i.d have recently been shown to not matter so we can proceed nevertheless. Example: If we take a covariance matrix of all stock tickers beginning in A comparing the average daily return over a period of time (or any process with no underlying relation), with each entry we can assume lognormal distribution forming a, lets say, 10000 by 10000 symmetric matrix. Over a duration of time we get a sequence of such matrices. Each of the entries we assume to be i.i.d since the stocks have 'nothing' to do with each other (although a weaker results holds for non-i.i.d entries). Now this series of matrices forms a chain of covariance matrices tending towards the underlying covariance matrix of the entire history of the stocks (if we sampled correctly). Random Matrix Theory implications: We known from RMT that once we decompose these matrices into their eigenvalues, the eigenvalues tend to the semicircle distribution. Since this distribution is continuous, there's a spread in results i.e. there is some underlying variance to the eigenvalue decomposition of covariance matrices. Conclusion: Thus when we use covariance matrices shouldn't there be some sort of hypothesis test that's able to filter out this underlying distribution, similar to comparing normal distributions where we need to account for variance when comparing two mean values. This would be dependent on how i.i.d the random variables are, the size of the matrix, the number of samples taken, and the mean/variance of the random variables themselves. What would be weird about this hypothesis test is that we'd expects as $n$ gets larger so does the error bound, captured by the asymptotic relationship between the size and convergence to the semicircle distribution. TLDR: Is there some sort of hypothesis testing for PCA, or any eigenvalue method, that filters out the underlying tendency of random matrices to share similar eigen-distributions? Similar to how when you compare the mean of two normal distributions you need to perform a hypothesis test to account for the variance. Edit - Solution: I found this paper  which seems quite promising although it'll take a few days for me to fully understand. It seems like this is an on-going area of research. Any further comments or recommendations are always appreciated.
65962
I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped.   1) $\delta_{i\,j}\delta_{i\,j}$  2) $\delta_{i\,j}  \epsilon_{i\,j\,k}$  I have no idea how to approach evaluating these properties. Without knowing i, j, or k, how would I approach? I don't feel confident using the notation further until I can understand these properties.
977
Whenever I needed to find the inverse of a matrix, I was told to check if its determinant is not zero. However, once I directly applied the Gauss-Jordan's method for finding the inverse of matrix whose determinant was zero. The inverse matrix that I got looked pretty normal like any other (if there wasn't a mistake).  I want to know how does the determinant of the matrix is related to inverse of matrix or why is that if determinant is zero then inverse doesn't exist? What exactly is inverse?
102335
$$     \begin{bmatrix}     -2 &amp; -2 &amp; 0 &amp; 0 \\     -b &amp; -b &amp; 0 &amp; 0 \\      0 &amp; 0 &amp; -4 &amp; 4 \\      0 &amp; 0 &amp; b &amp; -b \\     \end{bmatrix} $$  I have this matrix and I need to find the values for $b$ that this matrix won't  be diagonalizable.  For what I got this matrix is diagonalizable for every $b$ value.  Please write me also the final answer, thank you very much.
92502
As I understand, we can get correlation by normalizing covariance using the equation  $$\rho_{i,j}=\frac{cov(X_i, X_j)}{\sigma_i \sigma_j}$$  where $\sigma_i=\sqrt{E[(X_i-\mu_i)^2]}$ is the standard deviation of $X_i$.  My concern is what if the standard deviation equals zero? Is there any condition that guarantees it cannot be zero?  Thanks.
90127
I am interested in describing the group of special orthogonal matrices $SO(n)$ by a set of parameters, in any dimension. I would also like to obtain an expression of the density of the Haar measure in this set of parameters. Could anyone help me on this or indicate a good reference? Thanks
114422
I am interested in whether or not a "correlation" of three variables is something, and if what, what would this be?  Pearson product moment correlation coefficient  $$\frac{\mathrm{E}\{(X-\mu_X)(Y-\mu_Y)\}}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$$  Now the question for 3 variables: Is    $$\frac{\mathrm{E}\{(X-\mu_X)(Y-\mu_Y)(Z-\mu_Z)\}} {\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)\mathrm{Var}(Z)}}$$  anything?  In R it seems like something interpretable:  &gt; a &lt;- rnorm(100); b &lt;- rnorm(100); c &lt;- rnorm(100) &gt; mean((a-mean(a)) * (b-mean(b)) * (c-mean(c))) / (sd(a) * sd(b) * sd(c)) [1] -0.3476942   We normally look at the correlation between 2 variables given a fixed third variable's value. Could someone clarify?
63384
I am interested in the geometric meaning of the multiple correlation $R$ and coefficient of determination $R^2$ in the regression $y_i = \beta_1 + \beta_2 x_{2,i} + \dots + \beta_k x_{k,i} + \epsilon_i $, or in vector notation,  $$\mathbf{y} = \mathbf{X \beta} + \mathbf{\epsilon}$$  Here the design matrix $\mathbf{X}$ has $n$ rows and $k$ columns, of which the first is $\mathbf{x}_1 = \mathbf{1}_n$, a vector of 1s that corresponds to the intercept $\beta_1$.   The geometry is more interesting in the $n$-dimensional subject space rather than in the $k$-dimensional variable space. Define the hat matrix:  $$\mathbf{H} = \mathbf{X \left(X^\top X \right)}^{-1} \mathbf{X}^\top$$  This is an orthogonal projection onto the column space of $\mathbf{X}$, i.e. the   through the origin spanned by the $k$ vectors representing each variable $\mathbf{x}_i$, the first of which is $\mathbf{1}_n$. Then $\mathbf{H}$ projects the vector of observed responses $\mathbf{y}$ onto its "shadow" on the flat, the vector of fitted values $\mathbf{\hat{y}} = \mathbf{Hy}$, and if we look along the path of the projection we see the vector of residuals $\mathbf{e} = \mathbf{y} - \mathbf{\hat{y}}$ forms the third side of a triangle. This should furnish us with two routes to a geometric interpretation of $R^2$:   The square of the multiple correlation coefficient, $R$, which is defined as the correlation between $\mathbf{y}$ and $\mathbf{\hat{y}}$. This will appear geometrically as the cosine of an angle. In terms of lengths of vectors: for instance $SS_\text{residual} = \sum_{i=1}^{n}e_i^2 = \|\mathbf{e}\|^2$.   I would be delighted to see a brief account which explains:   The finer details for (1) and (2), Why (1) and (2) are equivalent, Briefly, how the geometric insight lets us visualise basic properties of $R^2$, for instance why it goes to 1 when the noise variance goes to 0. (After all, if we can't intuit from our visualisation then it's no more than a pretty picture.)   I appreciate this is more straightforward if the variables are centred first, which removes the intercept from the question. However, in most textbook accounts which introduce multiple regression, the design matrix $\mathbf{X}$ is as I laid out. Of course it's fine if an exposition delves into the space spanned by the centred variables, but for insight into the textbook linear algebra, it would be very helpful to relate this back to what's happening geometrically in the uncentred situation. A really insightful answer might explain what exactly is breaking down geometrically  - i.e. when the vector $\mathbf{1}_n$ is removed from the spanning set. I don't think this last point can be addressed by considering the centred variables alone.
89441
Given the following matrix, I was asked to tell whether the matrix is the square of some matrix or not.  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; -1 &amp; 0\\ 0 &amp; 0 &amp; -1 \end{pmatrix} $$  Since this matrix is diagonal, a square root is  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; i &amp; 0\\ 0 &amp; 0 &amp; i \end{pmatrix} $$ Am I right?
34198
I have an expression for a covariance matrix $C$ in terms of the indices $i$ and $j$.  In this way I can analytically calculate the elements of my covariance matrix, however when I try to invert $C$ matlab gives a warning about the matrix being close to singular.  The inversion therefore doesn't work, by which I mean the multiplying $C$ by the inverse given i do not get the identity.  I have tried calculating the pseudo inverse but this also does not work.  I have also tried adding a small constant along the diagonal but again the results do not work.  In general I'm working with matrices of dimension 1200 but I will give low dimensional example matrix that has the same properties, i.e. the matrices are symmetric about the diagonal and the anti-diagonal     19.9939  19.9954  19.9958  19.9951  19.9933  19.9905  19.9865    19.9954  19.9973  19.9981  19.9978  19.9965  19.9940  19.9905    19.9958  19.9981  19.9993  19.9995  19.9985  19.9965  19.9933    19.9951  19.9978  19.9995  20.0000  19.9995  19.9978  19.9951    19.9933  19.9965  19.9985  19.9995  19.9993  19.9981  19.9958    19.9905  19.9940  19.9965  19.9978  19.9981  19.9973  19.9954    19.9865  19.9905  19.9933  19.9951  19.9958  19.9954  19.9939   As mentioned in the title the matrix isn't positive, however the the negative eigenvalues are very small suggesting that the matrix is not positive only due to machine precision.  The negative eigenvalues are $-1.4048e-14$ and $-2.4571e-15$.  How can I go about inverting these matrices?
79716
Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X,Y)=0$, if and only if $X$ and $Y$ are independent.      How to show that: $$\rho(X,Y)= \frac{\mu(XY)\,\mu(X^cY^c)-\mu(XY^c)\,\mu(X^cY)}{(\mu(X) \,\mu(X^c)-\mu(Y) \,\mu(Y^c))^{1/2}};0&lt;\mu(X)&lt;1;\,0&lt;\mu(Y)&lt;1$$
70289
I have a question on my assignment asking me to prove that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$. What I did so far is:  We know that $A\cdot A^t = I$ (since it is orthogonal), and that would mean that if $A$ is a matrix with elements $a,b,c,d$ then :  \begin{cases}a^2 + b^2 = 1\\ ac + bd = 0\\ c^2 + d^2 = 1\end{cases}  which implies that \begin{align}\det(A)&amp;= ad - bc\\ &amp;= -\frac{bd^2}{c} - cb\\ &amp;= -\frac{bd^2}{c} - \frac{c^2b}{c}\\ &amp;= -\frac{bd^2 + bc^2}{c}\\ &amp;= -\frac{b(d^2 + c^2)}{c}\\ &amp;= -\frac{b(1)}{c}\\ &amp;= -b/c\end{align}  I might be doing something wrong here, could someone please help me out understand this?
59862
I am currently struggling with an exercise where I have to treat a Block diagonal matrix (so it is a square matrix, where square block matrices are down the diagonal). Now I was wondering whether we can say something about the characteristic or minimal polynomial of the whole matrix, if we know the characteristic and minimal polynomial of the blocks? Anything you know could be helpful!
104673
I wonder how to build a 1-1 correspondence between [0,1] and [0,1). My professor offers an example such that 1 in the first set corresponds to 1/2 in the second set, and 1/2 in the first set corresponds to 1/4 in the second. I don't quite understand it. Does it mean every element in the first set corresponds to its half value in the second set? wouldn't that make some elements left out in the second set? Does it still count as 1-1 correspondence? Does it connect to Schroder-Bernstein theorem?