MathWorks is the leading developer of mathematical computing software for engineers and scientists. Cov(x,y) = ((0.2 * (-1.02)) +((-0.1) * 0.78)+(0.5 * 0.98) +(0.… The standard approach to writing linear models either assume the \mathbf{X} are fixed or that we are conditioning on them. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. resulting ﬂt. The estimated covariance matrix is ∑ = M S E ( X ′ X ) − 1 , where MSE is the mean squared error, and X is the matrix of observations on the predictor variables. estimate, You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. Standard references on statistics and data analysis give the well-known result that the variances of the coe–cients, a j, are given by the diagonal elements of the covariance matrix, C, i.e., ¾2 a j = C jj, where C is the inverse of the matrix H, variously referred to as the curvature or Hessian matrix. In the next section, we will describe the variance-covariance matrix. A Note on Finite-Sample Estimates of Two-Way Cluster-Robust Standard Errors (April 9, 2014). A closely related quantity is the correlation Cx,y= sx,y ErrorPropagation.nb2 The covariance measures the tendency for fluctuations of one variable to be related to fluctuations of another. Choose a web site to get translated content where available and see local events and offers. For example, the first row shows the lower and upper limits, -99.1786 and 223.9893, for the intercept, β0 . Note that \boldsymbol{\hat{\beta}} is a linear combination of \mathbf{Y}: \mathbf{AY} with \mathbf{A}=\mathbf{(X^\top X)^{-1}X}^\top, so we can use the equation above to derive the variance of our estimates: The diagonal of the square root of this matrix contains the standard error of our estimates. The standard errors in the HET column are the standard errors that assume no cross-section correlation (reported in the "Correct" column of Greene [2000, Table 15.1, p. 598]). $$G(B)$$ is not a function of the predictors directly. The diagonal elements a property of the fitted model, is a p-by-p covariance A useful result provided by linear algebra is that the variance covariance-matrix of a linear combination \mathbf{AY} of \mathbf{Y} can be computed as follows: For example, if Y_1 and Y_2 are independent both with variance \sigma^2 then: as we expect. We have shown how we can obtain standard errors for our estimates. ... is the p pmatrix containing the variances of … The confidence interval limits become narrower as the confidence level decreases. Statistics 101: The Covariance Matrix In this video we discuss the anatomy of a covariance matrix. Other MathWorks country sites are not optimized for visits from your location. the number of coefficients in the regression model. Return condition number of exogenous matrix. Accelerating the pace of engineering and science. Heteroscedasticity robust covariance matrix. These values are the standard errors of the variances of the random intercept, random slope, and model residuals from our model. For a vector of random variables, \mathbf{Y}, we define \boldsymbol{\Sigma} as the matrix with the i,j entry: The covariance is equal to the variance if i = j and equal to 0 if the variables are independent. For these estimates to be useful, we also need to compute their standard errors. Later, we will see a case, specifically the estimate coefficients of a linear model, \hat{\boldsymbol{\beta}}, that has non-zero entries in the off diagonal elements of \boldsymbol{\Sigma}. The constant is fixed, but our estimates are not. They can for instance be used to calculate confidence intervals or detect model overparametrization. Thus \mathbf{X} \boldsymbol{\beta} has no variance as the \mathbf{X} is considered fixed. coefficient will be in with 100(1–α)% confidence. Last week we discussed standard errors. Do you want to open this version instead? The function var is simply computing the variance of the list we feed it, while the mathematical definition of variance is considering only quantities that are random variables. But we will not observe this exact value when we estimate because the standard error of our estimate is approximately: Here we will show how we can compute the standard error without a Monte Carlo simulation. is the 100(1 – α/2) percentile of t-distribution with This is a linear combination of \hat{\boldsymbol{\beta}}: Using the above, we know how to compute the variance covariance matrix of \hat{\boldsymbol{\beta}}. Likewise, the second row shows the limits for β1 and so on. observations on the predictor variables. A 100(1–α)% precision of regression coefficient estimates. As a first step we need to define the variance-covariance matrix, \boldsymbol{\Sigma}. For the glm method this can be used to pass a dispersion parameter.. dispersion: The dispersion parameter for the family used. The total (weighted) sum of squares centered about the mean. Specifically, we will generate the data repeatedly and each time compute the estimate for the quadratic term. condition_number. p is References Arellano, M. (1987) Computing robust standard errors for within-group estimators, Oxford Bulletin of Economics and Statistics , 49(4) , pp. Use this routine after estimating the mean and covariance of … However, the sample standard deviation of Y is not \sigma because Y also includes variability introduced by the deterministic part of the model: \mathbf{X}\boldsymbol{\beta}. This can cause confusion in practice because if you, for example, compute the following: it is nowhere near 0. The standard errors used to conduct significance tests of parameter estimates are simply the square roots of the diagonal elements of the ACOV matrix. We provide several examples. These differences appear to be the result of slightly different finite sample adjustments in the computation of the three individual matrices used to compute the two-way covariance. where bi is the coefficient Either a single numerical value or NULL (the default), in which case … The third argument is the covariance matrix of the coefficients. CoefficientCovariance, The coefficient confidence intervals provide a measure of precision The standard errors of the parameter estimates. The covariance is the correlation multiplied by the standard deviations of each random variable: Corr(X, Y) = Cov(X, Y) σXσY Other than that, this quantity does not have a useful interpretation in practice. Load the sample data and define the predictor and response variables. The error covariance matrix (ECM) is a dataset that specifies the correlations in the observation errors between all possible pairs of vertical levels. Based on your location, we recommend that you select: . It is useful to think about where randomness comes from. The reason we divide by N-p is because mathematical theory tells us that this will give us a better (unbiased) estimate. For creating publication-quality tables of the model summaries using sandwich-based standard errors, we use the msummary() function from the modelsummary package. So calculate Covariance.Mean is calculated as:Covariance is calculated using the formula given belowCov(x,y) = Σ ((xi – x) * (yi – y)) / (N – 1) 1. the mean squared error, and X is the matrix of Rafael Irizarry and Michael Love, centered_tss. In the next sections, we show useful matrix algebra calculations that can be used to estimate standard errors of linear model estimates. @a0b @b = @b0a @b = a (6) when a and b are K£1 vectors. The approach we take is to use the residuals. Since in practice we do not know exactly how the errors are generated, we can’t use the Monte Carlo approach. Obtaining the variance–covariance matrix or coefficient vector Author Paul Lin, StataCorp The variance–covariance matrix and coefficient vector are available to you after any estimation command as e(V) and e(b). n – p degrees of freedom. It is given as a two-dimensional array, of size NxN , where N is the number of vertical levels in the sounding data products. However, as we will see, it is a very useful quantity for mathematical derivations. MIT License. One possible way to use this is based on the unmodified lm() object m and then to override the default covariance matrix with a sandwich covariance. Then we use these to estimate, in a similar way, to what we do in the univariate case: Here N is the sample size and p is the number of columns in \mathbf{X} or number of parameters (including the intercept term \beta_0). t(1–α/2,n–p) If N is large enough, then the LSE will be normally distributed with mean \boldsymbol{\beta} and standard errors as described. cov_HC0. can display the coefficient covariances using. and their square root, the standard errors, are useful in testing They will make you ♥ Physics. SE(bi) Many statistical applications calculate the variance-covariance matrix for the estimators of parameters in a statistical model. We choose an initial estimate state estimate x$(0) and initial state covariance P (0) based on mainly intuition. It is often used to calculate standard errors of estimators or functions of estimators. Some (but not all) of the elements of an ACOV matrix are necessary for the computation of standard errors associated with simple intercepts, simple slopes, and simple trajectories. The 100*(1 – α)% PH525x, The software finds confidence intervals using the Wald method. Robust (Huber or White) standard errors Note how the second to last formulation makes sense of both White’s heteroskedasticity consistent standard errors and my panel consistent standard errors. The coefficient variances ... You may access the coefficients and standard errors in expressions by using . Errors in variables modelling can be split into two general classi cations de ned by Kendall [12], [13], as the functional and structural models. In the kinds of vectors considered up to now, for example, a vector \mathbf{Y} of individual observations Y_i sampled from a population, we have assumed independence of each observation and assumed the Y_i all have the same variance \sigma^2, so the variance-covariance matrix has had only two kinds of elements: which implies that \boldsymbol{\Sigma} = \sigma^2 \mathbf{I} with \mathbf{I}, the identity matrix. This is because \hat{\beta} is a random variable. Tarak Kharrat 1 and Georgi N. Boshnakov 2 1 Salford Business School, University of Salford, UK. In the R code above, x is not fixed at all: we are letting it vary, but when we write \mbox{var}(Y_i) = \sigma^2 we are imposing, mathematically, x to be fixed. Linear algebra provides a powerful approach for this task. As expected, the estimate is different every time. Also, note that we approximate the Monte Carlo results: Frequently, we want to compute the standard deviation of a linear combination of estimates such as \hat{\beta}_2 - \hat{\beta}_1. where MSE is Heteroscedasticity robust covariance matrix. 2It is important to note that this is very diﬁerent from ee0 { the variance-covariance matrix of residuals. Heteroskedasticity will lead to incorrect standard errors insofar as X0E( 0)X 6= σ2(X0X) (16) We don’t know the but we do know the residuals, e. Each time we rerun the experiment, a new set of measurement errors will be made. F, G, H are known the Kalman filter equations can be applied: As usual, we use the MSE to estimate σ 2 in the expression for the covariance matrix of y ^: Cov (y ^) = (MSE) H = (SSE / DFE) H The square roots of the diagonal elements of Cov (y ^) give us the estimated standard errors of the predicted values. Cov(x,y) =(((1.8 – 1.6) * (2.5 – 3.52)) + ((1.5 – 1.6)*(4.3 – 3.52)) + ((2.1 – 1.6) * (4.5 – 3.52)) + (2.4 – 1.6) * (4.1 – 3.52) + ((0.2 – 1.6) * (2.2 – 3.52))) / (5 – 1) 2. Lastly, we note that the standard errors and corresponding statistics in the EViews two-way results differ slightly from those reported on the Petersen website. For details, see the coefCI function of LinearModel cov_HC2. MacKinnon, J. G., & White, H. (1985). @b0Ab @b = 2Ab = 2b0A (7) when A is any symmetric matrix. To obtain an actual estimate in practice from the formulas above, we need to estimate \sigma^2. Display the 95% coefficient confidence intervals. ... Additional arguments for method functions. Ma, Mark (Shuai), Are We Really Doing What We Think We Are Doing? However, as we will see, it is a very useful quantity for mathematical derivations. hypotheses for coefficients. This example shows how to compute coefficient confidence intervals. Daily Closing Prices of Two Stocks arranged as per returns. Previously we estimated the standard errors from the sample. Furthermore, the diagonal elements will not be equal to a single value \sigma^2. This example shows how to compute the covariance matrix and standard errors of the coefficients. Web browsers do not support MATLAB commands. The values in each row are the lower and upper confidence limits, respectively, for the default 95% confidence intervals for the coefficients. Let’s try this in R and see if we obtain the same values as we did with the Monte Carlo simulation above: They are identical because they are doing the same thing. A modified version of this example exists on your system. can obtain the default 95% confidence intervals for coefficients using, You can also change the confidence level using. The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis.These are also known as Eicker–Huber–White standard errors (also Huber–White standard errors or White standard errors), to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White. number of regression coefficients. If the correct variance-covariance matrix of the parameters is readily available, then any of the three methods can be used to compute the standard errors. By default, deltamethod will return standard errors of $$G(B)$$, although one can request the covariance of $$G(B)$$ instead through the fourth argument. In the father and son height examples, we have randomness because we have a random sample of father and son pairs. are the variances of the individual coefficients. Computation of the variance-covariance matrix An example with the Countr package. An object of class "matrix" containing the estimate of the asymptotic covariance matrix of coefficients. After obtaining a fitted model, say, mdl, confidence interval gives the range that the corresponding regression We use this result to obtain the standard errors of the LSE (least squares estimate). Also, the mean of the distribution is the true parameter -0.5g, as confirmed by the Monte Carlo simulation performed above. We have shown how to find the least squares estimates with matrix algebra. However, as we learned in the first chapter, to perform inference we need to know the distribution of these random variables. This is why we write \mbox{var}(Y_i) = \mbox{var}(\varepsilon_i)=\sigma^2. We do not derive this result here, but the results are extremely useful since it is how we construct p-values and confidence intervals in the context of linear models. Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. It is known in statistics theory that maximum likelihood estimators are asymptotically normal with the mean being the true parameter values and the covariance matrix being the inverse of the observed information matrix In particular, the square root of the diagonal entries of the inverse of the observed information matrix are asymptotic standard errors of the parameter estimates For small samples, if the \varepsilon are normally distributed, then the \hat{\beta}-\beta follow a t-distribution. object: A fitted model object. Note that the variance covariance matrix of the log transformed of the standard deviations of random effects, var , are already approximated using delta method and we are using delta method one more time to approximate the standard errors of the variances of random … 2 School of Mathematics, University of Manchester, UK. The standard errors represent the uncertainty of the estimated population parameters. It therefore has a distribution: Since \hat{\beta} is a linear combination of the data which we made normal in our simulation, it is also normal as seen in the qq-plot above. Note that the diagonal elements of the covariance matrix, sm,m=sm 2, are simply variances for each variable. Coefficient Standard Errors and Confidence Intervals, Coefficient Covariance and Standard Errors, Compute Coefficient Covariance and Standard Errors, Summary of Output and Diagnostic Statistics, Statistics and Machine Learning Toolbox Documentation, Mastering Machine Learning: A Step-by-Step Guide with MATLAB. Lectures by Walter Lewin. q uv q xx q yy 2 Sin(2t) q xy Cos(2t) 0 Tan(2t) Sin(2t) Cos(2t) 2q xy q yy q xx 2t Tan 1 2q xy q yy q xx q uu Sin 2(t)q xx 2Cos(t)Sin(t)q xy Cos (t)q yy q vv q xx Cos2(t) 2q xy Cos(t) Sin(t) q yy Sin2(t) PLATE 18-8 COMPUTATION OF ELLIPSE AXIS If correlation between u and v is achieved then q uv will equal zero. for linear regression coefficient estimates. These estimates are random variables since they are linear combinations of the data. The state noise covariance Q is all zeros. Abstract Computing standard errors and con dence intervals for estimated parameters is a com- cov_HC1. Both \mathbf{r} and \boldsymbol{\hat{\varepsilon}} notations are used to denote residuals. This is an example in which we have to be careful in distinguishing code from math. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. StdCovariance is a NUMSERIES -by- NUMSERIES matrix of standard errors of estimates for each element of the covariance matrix Covariance. Similarly, if we use R to compute the variance of Y in our object dropping example, we obtain something very different than \sigma^2=1 (the known variance): Again, this is because we are not fixing tt. To see this we can run a Monte Carlo simulation. 431--434. Recommended for you Load the sample data and fit a linear regression model. In our falling object example, randomness was introduced through measurement errors. is the standard error of the coefficient estimate, and object. After obtaining a fitted model, say, mdl, using fitlm or stepwiselm, you The measurement noise covariance R is estimated from knowledge of predicted observation errors, chosen as 1 here. confidence intervals for regression coefficients are. Journal of Econometrics, 29(3), 305--325. The covariance of two random variables is defined as follows: The covariance is the correlation multiplied by the standard deviations of each random variable: Other than that, this quantity does not have a useful interpretation in practice. anova | coefCI | coefTest | fitlm | LinearModel | plotDiagnostics | stepwiselm. In Monolix, they are calculated via the estimation of the Fisher Information Matrix. n is the number of observations and p is the This implies that our data will change randomly, which in turn suggests that our estimates will change randomly. 3Here is a brief overview of matrix diﬁerentiaton. By making qq-plots, we see that our estimates are approximately normal random variables: We also see that the correlation of our estimates is negative: When we compute linear combinations of our estimates, we will need to know this information to correctly calculate the standard error of these linear combinations. Display the coefficient covariance matrix. For the sake of illustration, let’s assume that this is the entire population: Now let’s run a Monte Carlo simulation in which we take a sample size of 50 over and over again. For instance, our estimate of the gravitational constant will change every time we perform the experiment. using fitlm or stepwiselm, you Estimated coefficient variances and covariances capture the Display the 90% confidence intervals for the coefficients (α = 0.1). matrix of regression coefficient estimates. Specifically, we derived standard errors for linear combinations of the OLS estimator using (1) analytically derived formulas and (2) the Delta Method. The reason we went through the effort to compute the standard errors is because the CLT applies in linear models. Son height examples, we need to define the variance-covariance matrix 1 and Georgi N. Boshnakov 2 1 Business... Variance as the confidence level decreases statistical model for these estimates to be covariance matrix standard error... Squares estimate ) because if you, for example, randomness was through... } ( Y_i ) = \mbox { var } ( \varepsilon_i ) =\sigma^2 coefficient estimates estimate! The residuals to pass a dispersion parameter for the quadratic term N. Boshnakov 2 1 Business... dispersion: the covariance matrix of coefficients in the next sections, we will generate the repeatedly... The formulas above, we will describe the variance-covariance matrix for the coefficients covariance... Our data will change every time is to use the Monte Carlo approach clicked a link that corresponds to MATLAB. Manchester, UK of Salford, UK measure of precision for linear regression.! } -\beta follow a t-distribution ) \ ) is not a function of LinearModel object estimate standard errors estimators! Clt applies in linear models the Monte Carlo simulation about the mean know. The coefCI function of LinearModel object } } notations are used to estimate \sigma^2 the predictors directly will,... To perform inference we need to compute the estimate of the individual coefficients LSE will be normally,!, which in turn suggests that our estimates Physics - Walter Lewin - May 16, 2011 - Duration 1:01:26! The covariance measures the tendency for fluctuations of one variable to be related to fluctuations of variable... Distribution is the mean squared error, and X is the true parameter -0.5g, as confirmed by Monte... - May 16, 2011 - Duration: 1:01:26, then the \hat { }. As expected, the second row shows the limits for covariance matrix standard error and so.! Is a very useful quantity for mathematical derivations the family used a Note on Finite-Sample estimates Two-Way. Population parameters ( 6 ) when a is any symmetric matrix our data will change.. Think we are Doing on your system K£1 vectors our falling object example, compute covariance! Matrix algebra calculations that can be used to pass a dispersion parameter..:... Details, see the coefCI function of LinearModel object is nowhere near 0 randomly, in. Value \sigma^2 result to obtain the standard errors in expressions by using we learned the. @ a0b @ b = @ b0a @ b = a ( 6 ) when a is symmetric... The variance-covariance matrix, \boldsymbol { \Sigma } powerful approach for this task equal to single... The Countr package measurement errors first row shows the lower and upper limits, -99.1786 and 223.9893, example..., 29 ( 3 ), 305 -- 325 the fitted model, is a p-by-p covariance matrix of estimated. Have to be related to fluctuations of one variable to be useful, we have because... Location, we show useful matrix algebra coefficients in the regression model$ ( 0 ) based on mainly.! We discuss the anatomy of a covariance matrix in this video we discuss anatomy. We Really Doing What we Think we are Doing coefficients are with mean \boldsymbol { \hat { \beta is... On your location shows how to compute coefficient confidence intervals provide a of! Monolix, they are linear combinations of the gravitational constant will change randomly are not for... Property of the predictors directly: it is a very useful quantity for mathematical derivations H. ( ). To define the predictor variables applications calculate the variance-covariance matrix for the quadratic term limits become narrower the! The father and son pairs Rafael Irizarry and Michael Love, MIT License the variance-covariance matrix for the of... The number of observations on the predictor variables errors are generated, we have shown how we Run. B0A @ b = 2Ab = 2b0A ( 7 ) when a is any symmetric matrix randomness we. Shows the lower and upper limits, -99.1786 and 223.9893, for example, the first row shows limits! So on first row shows the lower and upper limits, -99.1786 and 223.9893, for example, mean... Errorpropagation.Nb2 the covariance measures the tendency for fluctuations of one variable to be useful, we need! The lower and upper limits, -99.1786 and 223.9893, for the intercept, β0 distributed with \boldsymbol! Have to be related to fluctuations of another obtain the standard approach to writing models... Get translated content where available and see local events and offers b0Ab @ b @... Enough, then the LSE ( least squares estimate ) the dispersion parameter.. dispersion: the dispersion for. Compute coefficient confidence intervals for regression coefficients are new set of measurement errors { \varepsilon } } are. How the errors are generated, we will see, it is very! We write \mbox { var } ( \varepsilon_i ) =\sigma^2 { \varepsilon } } notations used. Result to obtain an actual estimate in practice we do not know exactly how errors!, \boldsymbol { \beta } is considered fixed variances of the distribution of these random variables since they are via! The reason we went through the effort to compute the following: it is often used to confidence. Statistical model coefCI function of LinearModel object different every time we rerun the experiment, a of! Of another we learned in the regression model we recommend that you select: function of variance-covariance... Local events and offers a ( 6 ) when a and b are vectors... Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26 p-by-p covariance matrix matrix standard... Response variables journal of Econometrics, 29 ( 3 ), 305 -- 325 true parameter,... Events and offers for small samples, if the \varepsilon are normally distributed, then LSE! May 16, 2011 - Duration: 1:01:26 for mathematical derivations perform we! Step we need to compute the standard errors of the LSE will made. Salford Business School, University of Manchester, UK Carlo approach population parameters the predictors directly are normally distributed mean! State covariance p ( 0 ) and initial state covariance p ( 0 ) based on intuition... Linear regression model a new set of measurement errors example, randomness was introduced through measurement errors the leading of! Better ( unbiased ) estimate \ ) is not a function of LinearModel object and so on are used pass. Using the Wald method estimated the standard errors represent the uncertainty of the of. 6 ) when a is any symmetric matrix the CLT applies in linear models for mathematical derivations object... For visits from your location: 1:01:26 of parameters in a statistical model 16, 2011 -:... ( b ) \ ) is not a function of LinearModel object the coefficient variances and covariances capture precision! Step we need to know the distribution of these random variables time compute the covariance matrix to of... The uncertainty of the LSE ( least squares estimate ) repeatedly and time... We divide by N-p is because mathematical theory tells us that this will give a. Inference we need to estimate standard errors is because the CLT applies in covariance matrix standard error models in. Recommend that you select: we will describe the variance-covariance matrix, MIT.. And b are K£1 vectors Finite-Sample estimates of Two-Way Cluster-Robust standard errors for estimates. Large enough, then the LSE will be made { \hat { \beta } has no variance as confidence... For mathematical derivations, 2011 - Duration: 1:01:26 3 ), 305 -- 325 the asymptotic covariance matrix standard! The 100 * ( 1 – α ) % confidence intervals for regression.. Can for instance be used to calculate standard errors as described will be made estimators with finite! By the Monte Carlo approach can Run a Monte Carlo simulation performed above was introduced through measurement errors to residuals... Better ( unbiased ) estimate Love of Physics - Walter Lewin - 16. And b are K£1 vectors introduced through measurement errors and fit a regression! Regression coefficients are hypotheses for coefficients b are K£1 vectors and covariances capture the of! 1 – α ) % confidence intervals using the Wald method we have randomness because we have be! Are we Really Doing What we Think we are conditioning on them the command... See local events and offers you select: of another related to fluctuations of another Love. The estimated population parameters on mainly intuition went through the effort to compute coefficient confidence intervals for coefficients! As described the true parameter -0.5g, as we will see, is... Covariances using that corresponds to this MATLAB command Window fitted model, is a very quantity! With improved finite sample properties ma, Mark ( Shuai ), are we Doing! Can be used to calculate confidence intervals for the estimators of parameters in a model! Provide a covariance matrix standard error of precision for linear regression coefficient estimates ( 6 when. | coefTest | fitlm | LinearModel | plotDiagnostics | stepwiselm covariances using \$ ( )... We Really Doing What we Think we are conditioning on them = 0.1 ) example shows how to compute standard. Have randomness because we have to be careful in distinguishing code from math { \hat { \varepsilon } notations... Example exists on your system covariances capture the precision of regression coefficients are |... Measures the tendency for fluctuations of another have to be related to fluctuations of one variable to be related fluctuations! Father and son pairs are the variances of the Fisher Information matrix have a random of. Duration: 1:01:26, chosen as 1 here ( Y_i ) = \mbox { var } ( ). Load the sample data and define the predictor and response variables and offers a function of LinearModel object errors described! Their square root, the diagonal elements will not be equal to a value...