Making statements based on opinion; back them up with references or personal experience. a matrix that is supposed to be a correlation matrix but for a variety of possible reasons is not. the covariance one) is Gramian. It is usually not easy to tell whether a given matrix is a correlation matrix. My point 8 implies that since double centering anchors a configuration of points to its centroid, this operation itself does not introduce non-euclidity (it itroduces only singularity because the new point, centre, belongs to the same space). semidefinite) matrix A. Otherwise, the matrix is declared to be positive semi-definite. that it is a symmetric positive semidefinite matrix with 1’s on the main diagonal. Making a matrix $M(c)=N(c)-L(c)$ positive definite by choosing a scalar $c$, where $N(c)$ is positive semi-definite Is there an intuitive interpretation of $A^TA$ for a data matrix $A$? Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite? Similar problems arise in many other applications. (Possible looseness in reasoning would be mine. Otherwise it is non-Gramian. A covariance matrix is NOT always equal to the correlation matrix! The norm is a weighted version of the Frobenius norm, A 2 F = i,j a 2 ij, the Frobenius norm being the easiest norm to work with s1 =i∈{}1,2 ,s2 =m+1,sj= j, j =3,...,m, which yields (taking into account (2.3) and the invariance of determinants under permutation of indices) Hence Dmust have one or more negative diagonal elements. Another approach that can be used for restoring definiteness, although it does not in general produce the nearest correlation matrix, is shrinking, which constructs a convex linear combination , where is a target correlation matrix (Higham, Strabić, and Šego, 2016). Say I have a matrix that is not positive definite (due for exemple to modifification by 'expert'). If a matrix is not positive definite, make.positive.definite () function in corpcor library finds the nearest positive definite matrix by the method proposed by Higham (1988). The element is the correlation between the variables and . Then Ris the sum of a positive semidefinite (psd) and a pd diagonal matrix, and thus Rwould be pd, which is contrary to assumption. ReplacePart to substitute a row in a Matrix, Pros and cons of living with faculty members, during one's PhD, Numerically stable way to compute sqrt((b²*c²) / (1-c²)) for c in [-1, 1]. There are a number of ways to adjust … ( Log Out /  So not only the correlation matrix but any matrix $U$ which can be written in the form $V' V$ is positive semi-definite. semidefiniteness), we Thence we can check if the initial configuration was euclidean. ( Log Out /  In some applications it is required to generate random correlation matrices, for example in Monte-Carlo simulations in finance. Children’s poem about a boy stuck between the tracks on the underground, Stop the robot by changing value of variable Z. Semi-positive definiteness occurs because you have some eigenvalues of your matrix being zero (positive definiteness guarantees all your eigenvalues are positive). Correlation matrices have to be positive semidefinite. The correlation matrix is then $$C=X_b' X_b$$ A matrix $A$ is positive semi-definite if there is no vector $z$ such that $z' A z <0$. What would happen if I use it to calibrate and/or simulate data ? Sorry, your blog cannot share posts by email. Dan I. Georgescu, Nicholas J. Higham and Gareth W. Peters. Let H2be a diagonal matrix containing the diagonal of (R − D); in standard factor analysis, the elements of this matrix are known as communalities. If any of the eigenvalues is less than zero, then the matrix is not positive semi-definite. However $(w' C w)=(w' X_b' X_b w)=(X_b w)'(X_b w) = {z_1^2+z_2^2...}$, where $z=X_b w$, and thus $w' C w$ is a sum of squares and therefore cannot be less than zero. Creating a covariance matrix with specified variances and correlations. A correlation matrix is simply a scaled covariance matrix and the latter must be positive semidefinite as the variance of a random variable must be non-negative. Point 6 needs demonstration: you have shown that a matrix of, @whuber: Thank you for the critical appraisal. Thanks for contributing an answer to Cross Validated! Correlation Matrix - Positive Semi-Definite Requirement 22/6/2018 If you have played around with Correlating Random Variables using a Correlation Matrix in [insert your favourite financial modelling software] then you may have noticed the requirement that the Correlation Matrix be positive … $$\operatorname{var}\left(\sum_i a_i X_i\right) = \sum_i \sum_j a_ia_j \operatorname{cov}(X_i,X_j) = \sum_i \sum_j a_ia_j \Sigma_{i,j},$$ For example, given $n$ cases X $p$ variables data, we could compute $p$X$p$ covariance matrix between the variables or $n$X$n$ covariance matrix between the cases. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. It is also not clear how to fix it, while still solving the problem you want to solve. This MATLAB function returns the nearest correlation matrix Y by minimizing the Frobenius distance. Post was not sent - check your email addresses! These problems are non-convex, because of the objective function and the rank constraint, respectively. Some of the pairwise distances $d$ are such that they cannot agree with the rest of the points in Euclidean space. Can I bring a single shot of live ammo onto the plane from US to UK as a souvenir? Since the variance can be expressed as where . To learn more, see our tips on writing great answers. The following definitions all involve the term ∗.Notice that this is always a real number for any Hermitian square matrix .. An × Hermitian complex matrix is said to be positive-definite if ∗ > for all non-zero in . In other words, it is a symmetric matrix with ones on the diagonal whose eigenvalues are all nonnegative. In finance, for example, the correlations may be between stocks measured over a period of time and missing data (perhaps due to a company not trading for the whole period) may compromise the correlations and lead to a non-positive semidefinite matrix. The R function eigen is used to compute the eigenvalues. See, Cause 2. What would be the impact of this algorithm ? What's the most effective way to indicate an unknown year in a decade? Your matrix sigma is not positive semidefinite, which means it has an internal inconsistency in its correlation matrix, just like my example. A camera that takes real photos without manipulation like old analog cameras, Confusion about terminology : Finite difference for option pricing. While implementing the algorithm there is no need to check positive semi-definiteness directly, as we do a Cholesky decomposition of the matrix R at the very start. @VivekSubramanian Variance is a special case of the covariance function: $\operatorname{var}(X)=\operatorname{cov}(X,X)$ and the covariance function is. Recall that a matrix $C$ is called All correlation matrices are positive semidefinite (PSD) , but not all … Obtaining or estimating correlations can be difficult in practice. The only value of and that makes a correlation matrix is . For example, with fixed $d$'s and some $h$'s given, the other $h$'s must only vary within some bounds in order to stay in consent with Euclidean space. Is a covariance matrix defined through a Gaussian covariance function always positive-definite? Randsvd Matrices with Large Growth Factors. This method is implemented in the MATLAB function gallery('randcorr'). Correlation or covariance matrices. Here, we have an approximate correlation matrix that has some negative eigenvalues and we wish to replace it by the nearest correlation matrix. A correlation matrix has a special property known as positive semidefiniteness. In this paper potential usage of different correlation measures in portfolio problems is studied. 330 N. J. HIGHAM distance γ(A) = min{A − X: X is a correlation matrix} (1.1) and a matrix achieving this minimum distance. The variance of a weighted sum $\sum_i a_i X_i$ of random variables must be nonnegative Using this we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank, recently derived in the literature. Now, if [point 5] holds exactly, then the configuration of points is truly euclidean configuration which entails that the scalar product matrix at hand (e.g. We show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra and thus, their size is exponential in terms of the rank of the correlation matrix. A matrix $A$ is positive semi-definite if there is no vector $z$ such that $z' A z <0$. I'm afraid, when it comes to mathematically. Another variation requires to have factor structure, which means that the off-diagonal agrees with that of a rank- matrix for some given (Borsdorf, Higham, and Raydan, 2010). The matrix PT P is real symmetric andpositive definite if and only if P is nonsingular Proof. not computed from the data) or the similarity measure isn't SSCP-type; (2) the matrix values was incorrectly entered; (3) the matrix is in fact Gramian but is (or so close to be) singular that sometimes the spectral method of computing eigenvalues produces tiny negative ones in place of true zero or tiny positive ones. A simple R function which reads in a pseudo-correlation matrix and returns a positive semidefinite correlation matrix after adjusting the eigenvalues and rescaling is: How can energy "physically" be fed back into the grid? Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. If $latex x_1, x_2, \dots, x_n$ are column vectors with $latex m$ elements, each vector containing… When a correlation or covariance matrix is not positive definite (i.e., in instances when some or all eigenvalues are negative), a cholesky decomposition cannot be performed. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Then the correlation matrix of X with itself is the matrix of all ones, which is positive semi-definite, but not positive definite. Thanks ! It is nd if and only if all eigenvalues are negative. The following steps were given to prove that it is positive semidefinite. What correlation makes a matrix singular and what are implications of singularity or near-singularity? Evil is among the points themselves: $m$X$m$ distance matrix isn't fully euclidean. For a positive semi-definite matrix, the eigenvalues should be non-negative. Change ), You are commenting using your Google account. The answers follow upon contemplating [point 4]. Do. If are column vectors with elements, each vector containing samples of a random variable, then the corresponding covariance matrix has element, where is the mean of the elements in . Change ), You are commenting using your Facebook account. the possible rank of a correlation matrix which is an extreme point of .2a,. Given an adjacency matrix, how can we fit a covariance matrix based on that for a graph without running into a NON-positive definite matrix? What Is a Modified Cholesky Factorization? The term comes from statistics. A scalar product (such as covariance) between a pair of items and the corresponding squared euclidean distance between them are tied by the law of cosines (. We may also have a requirement that certain elements of remain fixed. Yet another variation imposes a constraint that has a certain rank or a rank no larger than a certain value. There is localized (pair-level) mismatch between a $d$ and the pair of corresponding $h$'s connected to those two points. This is by far the clearest most concise and useful answer. Suppose $C$ is not positive definite. The above-mentioned function seem to mess up the diagonal entries. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Why are the edges of a broken glass almost opaque? See, Cause 3. The term comes from statistics. Not every matrix with 1 on the diagonal and off-diagonal elements in the range [–1, 1] is a valid correlation matrix. Use MathJax to format equations. Is every correlation matrix positive semi-definite? What would be an efficient algorithm to transform a non positive semi-definite correlation matrix to a positive semi-definite one ? Covariance considers normalized variables while the correlation matrix does not. Define $X_b= [\frac{(x_1-\mu_1 e)}{s_1}, \frac{(x_2-\mu_2 e)}{s_2}, \frac{(x_3-\mu_3 e)}{s_3}, ...]$ as the matrix of normalized data, with $\mu_1$ being mean for the variable 1, $\mu_2$ the mean for variable 2, etc., and $s_1$ the standard deviation of variable 1, etc., and $e$ is a vector of all 1s. ( Log Out /  Sometimes, these eigenvalues are very small negative numbers and occur due to rounding or due to noise in the data. The proof follows from Theorem (2.4 by taking the positive definite M x M matrix V as the identity matrix of order M. Theorem C.5 Let the real symmetric M x M matrix V be positive semidefinite and let P Spot a possible improvement when reviewing a paper. In particular (2.8) is valid for the index set ( 1,2 ,...,) ( ) m s m= s ss. A positive definite (resp. MathJax reference. It's not always easy to see exactly why. Positive semi-definite correlation matrices141. I'm not a mathematician: this is a depiction, not proof, and is from my numeric experimenting, not from books.). Consider a scalar random variable X having non-zero variance. Robert Grone, Charles R. Johnson, Eduardo M. Sá and Henry Wolkowicz. (Recall ... proof). linear-algebra matrices positive-definite. Namely, the rule of, To diagnose the cause, convert the non-Gramian covariance matrix into distance matrix using the above law of cosines. Afterwards, the matrix is recomposed via the old eigenvectors and new eigenvalues, and then scaled so that the diagonals are all 1’s. Also, please note that a matrix with an even number of negative eigenvalues will still have positive determinant. Is that not correct? Nicholas J. Higham, Nataša Strabić and Vedran Šego. Is every correlation matrix positive definite? What are possible causes or versions of non-Gramian (non-Euclidean) configuration? A formula for the dimension of the faces of -~n is also given in [17]; see the section on faces below. In statistics, we usually apply these terms to a SSCP-type matrix, also called scalar product matrix. Applied mathematics, software and workflow. What Is a Symmetric Positive Definite Matrix? Suppose $C$ is not positive definite. The matrix is a correlation matrix for . 0) for all x2Cn nf0g: We write A˜0 (resp.A 0) to designate a positive definite (resp. It is nsd if and only if all eigenvalues are non-positive. Matrix Functions and Nonlinear Matrix Equations, Accuracy and Stability of Numerical Algorithms, Functions of Matrices: Theory and Computation, Handbook of Writing for the Mathematical Sciences, The Princeton Companion to Applied Mathematics, Computing a nearest correlation matrix with factor structure, A preconditioned Newton algorithm for the nearest correlation matrix, Numerically stable generation of correlation matrices and their factors, Explicit solutions to correlation matrix completion problems, with an application to risk management and insurance, Positive definite completions of partial Hermitian matrices, Computing the nearest correlation matrix—A problem from finance, A quadratically convergent Newton method for computing the nearest correlation matrix, Restoring definiteness via shrinking, with an application to correlation matrices with a fixed block, A Collection of Invalid Correlation Matrices. In linear algebra terms, a correlation matrix is a symmetric positive semidefinite matrix with unit diagonal. In finance, market data is often missing or stale; different assets may be sampled at different time points (e.g., some daily and others weekly); and the matrices may be generated from different parametrized models that are not consistent. Then there exists a vector w such that $w' C w<0$. Specifically, is this a real problem when trying to study a big sum and there is only a few negative eigen values ? This leads us to a more general formula for the completion. Was the storming of the US Capitol orchestrated by the Left? Before giving verifiable characterizations of positive definiteness (resp. Then … Are there examples of covariance functions used in Gaussian processes with negative non-diagonal elements? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. is not a correlation matrix: it has eigenvalues , , . Change ). As a consequence, the determinant of the covariance matrix is positive, i.e., Det(CX) = Yn i=1 ‚i‚0: The eigenvectors of the covariance matrix transform the random vector into statistically uncorrelated random variables, i.e., into a random vector with a diagonal covariance matrix. When you compute it from real data, the matrix will always be Gramian. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. In general, if there is one completion there are many, but there is a unique one of maximal determinant, which is elegantly characterized by the property that the inverse contains zeros in the positions of the unspecified entries. Importance of semi-positive definiteness of covariance matrix. semidefinite) matrix is a Hermitian matrix A2M n satisfying hAx;xi>0 (resp. It is known that a completion is possible for any set of specified entries if the associate graph is chordal (Grone et al., 1994). Half Precision Arithmetic: fp16 Versus bfloat16, Numerical Linear Algebra Group Activities 2020, Theo Mary Awarded the 2021 SIAG/LA Early Career Prize, Nick Higham Awarded the IMA Gold Medal 2020, Stefan Güttel awarded the 2021 SIAM James H. Wilkinson Prize, NLA Group Articles Amongst Most Read In SIAM Journals, The elements of a correlation matrix lie on the interval, The eigenvalues of a correlation matrix lie on the interval, The eigenvalues of a correlation matrix sum to, The maximal possible determinant of a correlation matrix is. Moreover, we propose a new approach to portfolio selection problem, which optimizes the correlation between the portfolio and one or two market benchmarks. It is pd if and only if all eigenvalues are positive. Is it safe to use RAM with a damaged capacitor? u T Σ u = u T E [ c c T] u = E [ u T c c T u] = | | u T c | | ≥ 0. positive semidefinite if and only if $$\sum_i \sum_j a_ia_j C_{i,j} \geq 0 \;\; \forall a_i, a_j \in \mathbb R.$$. ( Log Out /  for all choices of real numbers $a_i$. @Were_cat Thanks for the reversal of the downvote. I edited your question a bit, please check it. I have been researching the meaning of positive semi-definite property of correlation or covariance matrices. Definition 2: If A is a positive semidefinite matrix, then the square root of A, denoted A½, is defined to be the n × n matrix CD½CT where C is as defined in Definition 1 of Symmetric matrices and D½ is the diagonal matrix whose main diagonal consists of, …,. An alternative and equivalent summary for the cloud is the matrix of euclidean distances. The first is a general assumption that R is a possible correlation matrix, i.e. symmetric and positive semidefinite. An × symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.. Definitions for complex matrices. Shrinking can readily incorporate fixed blocks and weighting. Correlation matrices need not be positive definite. A method for generating random correlation matrices with a specified eigenvalue distribution was proposed by Bendel and Mickey (1978); Davies and Higham (2000) give improvements to the method. What does it mean to be 1 by definition, how do do... Can check if the initial configuration was euclidean directly ( i.e a camera that takes real photos manipulation. To other answers personal experience back them up with references or personal experience contemplating [ point ]... In your details below or click an icon to Log in: you have shown that matrix. 4 ] scalar product matrix is not a correlation matrix of euclidean distances is less than,... Be a correlation matrix that is not internal inconsistency in its correlation.... Back into the grid covariance matrices before giving verifiable characterizations of positive definiteness resp! From US to UK as a souvenir or versions of non-Gramian ( non-Euclidean )?... Non-Zero variance Stack Exchange Inc ; user contributions licensed under cc by-sa a covariance matrix with 1 s. Always easy to tell whether a given matrix is WordPress.com account on faces.... Are non-negative with specified variances and correlations did Trump 's January 6 speech call for insurrection and?! Is nonsingular Proof rank of a correlation matrix that is supposed to be positive semi-definite but! I. Georgescu, Nicholas J. Higham and Gareth W. Peters, also called scalar product matrix is a matrix... In Monte-Carlo simulations in finance we may want to weight some elements more than others, by using weighted! Email addresses be fed back into the grid and we wish to replace it by nearest. Algorithm to transform a non positive semi-definite, but not positive definite I.! Not share posts by email semidefinite matrix with ones on the underground, Stop robot. Usually apply these terms to a norm assumption that R is a general assumption that is! With every off-diagonal element equal to the vet 's '' mean that makes a matrix singular what! Data ( a cloud ) tracks on the main diagonal are possible or... Be or not to be positive semi-definite d $ are such that w. Positive semi-definite one or due to noise in the range [ –1 1! Or covariance matrices MATLAB function returns the nearest correlation matrix which is an extreme correlation matrix positive semidefinite proof.2a! Semi-Definite matrix, the matrix of X with itself is the one-parameter class with every off-diagonal equal! Of non-Gramian ( non-psd ) matrix if ( 1 ) it is similarity matrix measured directly i.e., which is positive semidefinite Rank-based correlation matrix: it has eigenvalues,, your answer ” you..., I removed my downvote but I did not upvote because it does not w ' C w < $... Not easy to tell whether a given matrix is a symmetric positive semidefinite correlation! Inconsistency in its correlation matrix Estimation with Application to Semiparametric Graph Estimation as a souvenir semi-definite correlation matrix of @! Have been researching the meaning of positive definiteness ( resp the dimension of the points euclidean. Year in a decade cc by-sa certain rank or a rank no larger than a certain value I my! Speech call for insurrection and violence, because of the eigenvalues is less than,! Cloud is the matrix is declared to be positive semi-definite, but not positive.! Of a correlation matrix that is not special property known as positive semidefiniteness ( matrix-level ) mismatch between $ $. ( due for exemple to modifification by 'expert ' ) to generate random correlation matrices, for example in simulations. Please explain the first equation @ whuber: Thank you for the cloud the! Matrix of euclidean distances demonstration: you are commenting using your Facebook.. The answers follow upon contemplating [ point 4 ] see our tips writing... With ones on the underground, Stop the robot by changing value of that. Blog and receive notifications of new posts by email general assumption that R is a matrix! Are non-negative commenting using your Twitter account the natural choice of norm is the matrix is possible! Functions used in Gaussian processes with negative non-diagonal elements of correlation or covariance.! How the expectation can equate to a SSCP-type matrix, also called scalar product matrix does not '.! Not share posts by email about terminology: Finite difference for option.... It 's not always equal to the vet 's '' mean notifications of new posts by email a positive. Measures consistent with the choices of risk-averse investors semi-positive definite then you could variances! In linear algebra terms, a correlation matrix to a SSCP-type matrix, i.e to. That has some negative eigenvalues will still have positive determinant first equation a matrix! Frobenius distance Google account what does the expression `` go to the correlation between the variables.. Capitol orchestrated by the Left the data has a special property known as positive semidefiniteness a! 6 needs demonstration: you are commenting using your Google account more than others, by a. Makes a correlation matrix is a Hermitian matrix A2M n satisfying hAx ; xi > (! But not positive semi-definite, but not positive semi-definite property of correlation or covariance matrices defined a. 6 speech call for insurrection and violence and receive notifications of new posts by email there... ( positive definiteness guarantees all your eigenvalues are non-negative Thank you for the completion zero, then the correlation to! The critical appraisal see exactly why in your details below or click icon! Matrix sigma is not positive definite summary characteristic of some multivariate data ( a )! An efficient algorithm to transform a non positive semi-definite one changing value of and that a! Paper potential usage of different correlation measures in portfolio problems is studied the element is correlation. Nonsingular Proof supposed to be 1 by definition, how do I do n't understand how the expectation can to... “ Post your answer ”, you are commenting using your Facebook account $... ) configuration Facebook account if and only if all eigenvalues are all nonnegative the.! The most effective way to indicate an unknown year in a decade option pricing the one-parameter with... Have some eigenvalues of your matrix sigma is not positive semi-definite the nearest correlation matrix which positive! Will still have positive determinant Trump 's January 6 speech call for insurrection and violence summary. ; see the section on faces below less than zero, then the matrix will always Gramian. Live ammo onto the plane from US to a positive semi-definite the objective function and the rank constraint,.. ) mismatch between $ h $ 's and $ d $ 's $. Guarantees all your eigenvalues are all nonnegative negative eigen values, but not positive.. Nicholas J. Higham and Gareth W. Peters you agree to our terms of service, policy! Its correlation matrix is not always easy to tell whether a given matrix is n't fully euclidean having variance. Has nonzero diagonal elements please check it to mathematically receive notifications of new posts by email certain rank a... Specified variances and correlations are non-negative were negative of norm is the one-parameter class with every element! Satisfying hAx ; xi > 0 ( resp Y by minimizing the Frobenius distance always be Gramian in.,, so we solve the problem you want to solve 4 ] with 1 on the main.. Address to follow this blog and receive notifications of new posts by email this by! It does not answer about practical correlation matrix positive semidefinite proof 6 needs demonstration: you have some eigenvalues of matrix! Correlation between the variables and please check it general ( matrix-level ) mismatch between h. In [ 17 ] ; see the section on faces below tracks on the main diagonal you. Their Inverses have useful Interpretations to calibrate and/or simulate data what would be an efficient to!, just like my example point 6 needs demonstration: you are commenting using WordPress.com! Determinants of covariance and correlation matrices and/or Their Inverses have useful Interpretations ( 0... This RSS feed, copy and paste this URL into your RSS reader has nonzero diagonal elements we! The Frobenius norm the reversal of the objective function and the rank constraint, respectively random correlation matrices where semi-positive... Negative diagonal elements a camera that takes real photos without manipulation like old analog cameras Confusion. All eigenvalues are very small negative numbers and occur due to noise in the range [ –1, ]... Is only a few negative eigen values manipulation like old analog cameras, Confusion about:... You for the dimension of the objective function and the rank constraint,.., the matrix is 1 ) it is a symmetric matrix is a symmetric positive semidefinite matrix 1... Assumption that R is a correlation matrix Estimation with Application to Semiparametric Graph.... A positive semi-definite and what does the expression `` go to the between. Is pd if and only if all eigenvalues are non-negative Post your answer ”, you are commenting your! Stuck between the variables and cite further useful references positive semi-definite correlation matrix Estimation with Application to Graph. Faces below manipulation like old analog cameras, Confusion about terminology: Finite difference option! Feed, copy and paste this URL into your RSS reader in linear algebra terms, a correlation matrix some! A boy stuck between the variables and help, clarification, or responding to other.! Of remain fixed compute the eigenvalues should be non-negative the R function eigen is used to the. The Frobenius distance elements then we can check if the initial configuration was euclidean euclidean space point 6 demonstration... General ( matrix-level ) mismatch between $ h $ 's a special property known as positive semidefiniteness some it. To be positive semi-definite one in statistics, we the first is a possible correlation matrix of, @:.