I think that if we write $H \mathbf{1}=\mathbf{b}$ and premultiply both sides by $X'$, we get $X'\mathbf{1}=X'\mathbf{b}=\mathbf{c}$, where $\mathbf{c}$ is some vector. These estimates will be approximately normal in general. Indeed, for any nxm matrix A, we have. 2.1 Residuals The vector of residuals, e, is just e y x b (42) Using the hat matrix, e = y Hy = (I H)y (43) Here are some properties of I H: 1. Introduction The ‘hat matrix’ plays a fundamental role in regression analysis; the elements of this matrix have well-known properties and are used to construct variances and covariances of the residuals. S.O.S. The only non-singular idempotent matrix is the identity matrix; that is, if a non ... (the latter being known as the hat matrix) are idempotent and symmetric matrices, a fact which allows simplification when the sum of squared residuals is computed: ^ ^ = () = = =. In this page, we give some general results about the three operations: addition, multiplication, and multiplication with numbers, called scalar multiplication. A commonly talked about property is the state or phase of matter. If you can perform the appropriate products, then we have Making statements based on opinion; back them up with references or personal experience. 3. Now that we have developed the hat matrix and a number of its properties, we turn to three examples, two designed and one sampled. These estimates are normal if Y is normal. The first species in the genus Homo evolved from the genus Australopithecus. Denoting $h_{ij}$ the typical element of $H$, the typical element of the first row of $X'H$ is, $$\left[X'H\right]_{1j} = \sum_{i=1}^n h_{ij} = \left[X'\right]_{1j}= 1 \;\;\forall j$$. Recall that H = [h ij]n i;j=1 and h ii = X i(X T X) 1XT i. I The diagonal elements h iiare calledleverages. There are currently five different states of matter that have been identified: solids, liquids, gases, plasma and Bose-Einstein … A vector field can be thought of as a set of vectors that all align in a particular direction in space (or more precisely, Euclidean space). We call this the \hat matrix" because is turns Y’s into Y^’s. Let 1 be the first column vector of the design matrix X. Hat Matrix Y^ = Xb Y^ = X(X0X)−1X0Y Y^ = HY where H= X(X0X)−1X0. Using properties of matrix operations Our mission is to provide a free, world-class education to anyone, anywhere. Using properties of matrix operations Our mission is to provide a free, world-class education to anyone, anywhere. Asking for help, clarification, or responding to other answers. Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 11, Slide 22 Residuals • The residuals, like the fitted values of \hat{Y_i} can be expressed as linear 1’=1H’=1’H & 1’H1=n; The elements of H are denoted by h ii i.e. Estimated Covariance Matrix of b This matrix b is a linear combination of the elements of Y. $$H_{n\times k}= X\left(X'X\right)^{-1}X' \Rightarrow X'H = X'$$, The first row of $X'$ is a row of ones, so $\left[X'\right]_{1j}=1$ . How many treble keys should I have for accordion? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Vector calculus plays a vital role in differential geometry and differential equations. The projection matrix has a number of useful algebraic properties. But $\left[X'H\right]_{1j}$ is the sum of the elements of the $j$ column of $H$, i.e. We have seen that matrix multiplication is different from normal multiplication (between numbers). Vector calculus can be defined as the branch of mathematics that deals with the differentiation and integration of vector fields. Please post your question on our Properties of Projection Matrices. Just note that yˆ = y −e = [I −M]y = Hy (31) where H = X(X0X)−1X0 (32) Greene calls this matrix P, but he is alone. I understand that the trace of the projection matrix (also known as the "hat" matrix) X*Inv(X'X)*X' in linear regression is equal to the rank of X. {\displaystyle {\hat {e}}^{\textsf {T}}{\hat {e}}=(My)^{\textsf {T}}(My)=y^{\textsf {T}}M^{\textsf {T}}My=y^{\textsf {T}}MMy=y^{\textsf {T}}My.} 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.. Visit Stack Exchange The hat matrix is a matrix used in regression analysis and analysis of variance.It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method. How can we prove that from first principles, i.e. How does the recent Chinese quantum supremacy claim compare with Google's? Formal Examples To illustrate the hat matrix and develop our intui-tion, we begin with two familiar examples in which Consider the linear regression model where the outputs are denoted by , the associated vectors of inputs are denoted by , the vector of regression coefficients is denoted by and are unobservable error terms. Stupid question: Why is the hat/projection matrix not the identity matrix? 1. The matrix 0 is the identity of matrix addition. Properties involving Addition. $QED$. B 2 = B. Let … Do I have to use any other property which I don't know? Is a password-protected stolen laptop safe? projection matrix for some subspace, but that’s also true. Thanks for contributing an answer to Cross Validated! So y ¼ Hy þMy ¼ ^yþe where, because of (3.11) and (3.13), ^y0e ¼ 0, so that the vectors ^y and e are orthogonal to each other. H1=1 with intercept term. The first humans evolved in Africa and later spread across the globe; replacing other hominid populations in areas they colonize, most notably neanderthals, which they interbred with (every human outside of Africa possess a tiny amount of the neanderthal genome). Let A, B, and C be mxn matrices. Note that it is possible to have two non-zero ma-trices which multiply to 0. With two standardized variables, our regression equation is . Therefore, when performing linear regression in the matrix form, if \( { \hat{\mathbf{Y}} } \) Your English is better than my <>. = 1 1 1 1 1 + 1 1 + 1! Let A, B, and C be three matrices. To learn more, see our tips on writing great answers. Hat Matrix: Properties and Interpretation Week 5, Lecture 1 1 Hat Matrix 1.1 From Observed to Fitted Values The OLS estimator was found to be given by the (p 1) vector, b= (XT X) 1XT y: The predicted values ybcan then be written as, by= X b= X(XT X) 1XT y =: Hy; where H := X(XT X) 1XT is an n nmatrix, which \puts the hat on y" and is therefore referred to as the My impression is that covering these properties is a holdover from the "New Math" fiasco of the 1960s. Mathematics CyberBoard. Obtaining b weights from a Correlation Matrix. The hat matrix H is defined in terms of the data matrix X: H = X (XTX) –1XT and determines the fitted or predicted values since We have. every row of H adds upto 1. How to give feedback that is not demotivating? Then the Toba eruptions had … Indeed, consider the nxn matrix, The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Are there some similarities? These properties of the hat matrix are of importance in, for example, assessing the amount of leverage or in uence that y j has on ^y i, which is related to the (i;j)-th entry of the hat matrix. ANOVA hat matrix is not a projection matrix, it shares many of the same geometric proper-ties as its parametric counterpart. In hindsight, it is … Is there a difference between a tie-breaker and a regular vote? In statistics, the projection matrix (), sometimes also called the influence matrix or hat matrix (), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). 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. up to date? Hat Matrix Y^ = Xb Y^ = X(X0X)−1X0Y Y^ = HY where H= X(X0X)−1X0. Example 4 1 1 1 1! Use MathJax to format equations. The Data Matrix Properties Calculating Variable (Column) Means The sample mean of the j-th variable is given by x j = 1 n Xn i=1 ij = n 110 nxj where 1n denotes an n 1 vector of ones xj denotes the j-th column of X Nathaniel E. Helwig (U of Minnesota) Data, Covariance, and Correlation Matrix Updated 16-Jan-2017 : … Do you need more help? It follows then that the trace (sum of diagonal elements - in this case sum of $1$ 's) will be the rank of the column space, while there'll be as many zeros as the dimension of the null space. Hence, the values in the diagonal of the hat matrix will be less than one (trace = sum eigenvalues), and an entry will be considered to have high leverage if … A related matrix is the hat matrix which makes yˆ, the predicted y out of y. Therefore, when performing linear regression in the matrix form, if Y ^ HX=X (I-H)X=0; HH=H 2 =H=H p; H(I-H)=0 $Cov(\hat{e},\hat{Y})=Cov\left\{HY,(I-H)Y\right\}=\sigma ^{2} H(I-H)=0$ 1-H is also symmetric and idempotent. We assume to observe a sample of realizations, so that the vector of all outputs is an vector, the design matrixis an matrix, and the vector of error termsis an vector. Formal Examples To illustrate the hat matrix and develop our intui-tion, we begin with two familiar examples in which 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.. Visit Stack Exchange We can translate the above properties of orthogonal projections into properties of the associated standard matrix. What to do? This matrix is symmetric (HT = H) and idempotent (HH = H) and is therefore a projection matrix; it performs the orthogonal projection of y on the K -dimensional subspace spanned by the columns of X. We then discuss (in Section 5) how to handle yi when hii indicates a high-leverage point. is called the hat matrix21 because it transforms the observed y into ŷ. I apologise for the utter ignorance of linear algebra in this post, but I just can't work it out. Hat Matrix and Leverages Basic idea: use the hat matrix to identify outliers in X. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Differential geometry is a mathematical discipline that uses the techniques o… Let H=[r1 r2 .. rn]', where rn is a row vector of H. I think prooving H1=H is the same as answering the question above. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Now that we have developed the hat matrix and a number of its properties, we turn to three examples, two designed and one sampled. 2.1 Residuals The vector of residuals, e, is just e y x b (42) Using the hat matrix, e = y Hy = (I H)y (43) Here are some properties of I H: 1. Some simple properties of the hat matrix are important in interpreting least squares. Properties involving Addition and Multiplication. How to whiten a white Ikea mattress cover? We call this the \hat matrix" because is turns Y’s into Y^’s. Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. Show that H1=1 for the multiple linear regression case(p-1>1). In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix $${\displaystyle \mathbf {X} }$$. We then discuss (in Section 5) how to handle yi when hii indicates a high-leverage point. where the n×n prediction or Hat matrix, H, is given by (1.4) H = X(X′X)−1X′. Estimated Covariance Matrix of b This matrix b is a linear combination of the elements of Y. Some facts of the projection matrix in this setting are summarized as follows: A Merge Sort implementation for efficiency. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now. The most important terms of H are the diagonal elements. From now on, we will not write (mxn) but mxn. 1 1 1 1! The residual vector is given by e = (In −H)y with the variance-covariance matrix V = (In −H)σ2, where In is the identity matrix of order n. The matrix H plays an important role in the linear regression analysis. My professor skipped me on christmas bonus payment. (Note that $${\displaystyle \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}}$$ is the pseudoinverse of X.) By writing H 2= HHout fully and cancelling we nd H = H. A matrix Hwith H2 = His called idempotent. The above DFT equation using the twiddle factor can also be written in matrix form. These estimates will be approximately normal in general. it is the inner product of this column with the vector of ones. It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method. is called the ‘hat matrix’, since it transforms y into ^y (pronounced: ‘y-hat’). HH = H Important idempotent matrix property For a symmetric and idempotent matrix A, rank(A) = trace(A), the number of non-zero eigenvalues of A. i.e. There are a few useful facts about the diagonal elements of the hat matrix: where K is the number of independent variables, including the constant if there is one. 2. Many of the basic properties of expected value of random variables have analogous results for expected value of random matrices, with matrix operation replacing the ordinary ones. $X'\mathbf{b}=\mathbf{c}$ has a unique solution if $X$ has full rank. Properties involving Addition and Multiplication. Windows 10 - Which services and Windows features and so on are unnecesary and can be safely disabled? is it possible to read and play a piece that's written in Gflat (6 flats) by substituting those for one sharp, thus in key G? We have a system of k +1 equations. I Properties of leverages h ii: 1 0 h ii 1 (can you show this? ) Hat Matrix Properties 1. the hat matrix is symmetric 2. the hat matrix is idempotent, i.e. How do we know $X'X$ is nonsingular in OLS? Let 1 be the first column vector of the design matrix X. Eventually they developed stone tools and primitive weaponry. Proving that $ (\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)$ is independent with SSE, Updating regression solutions for a new regressor without the original dependent variable, Updating regression solutions for removing a regressor without the original dependent variable. We will see later how to read o the dimension of the subspace from the properties of its projection matrix. It describes the influence each response value has on each fitted value. How late in the book-editing process can you change a characters name? But we must have full rank of $X$ to even make inverse of $X'X$. projection matrix for some subspace, but that’s also true. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Since the hat matrix is a projection matrix, its eigenvalues are $0$ and $1$. So if n is different from m, the two zero-matrices are different. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, a question on regression analysis ; property of Hat matrix. Mathematical Properties of Hat Matrix. We will see later how to read o the dimension of the subspace from the properties of its projection matrix. In uence @e i=@y j= (I H) ij. The matrix form of calculating a DFT and an IDFT eases up many calculations. And this hold for all columns of $H$. The ﬁrst order conditions are @RSS @ ˆ j = 0 ⇒ ∑n i=1 xij uˆi = 0; (j = 0; 1;:::;k) where ˆu is the residual. Clearly, there holds H0 ¼ H, H2 ¼ H, H þM ¼ I and HM ¼ 0. Properties Singularity and regularity. If X is the design matrix, then the hat matrix H is given by Tukey coined the term \hat matrix" for Hbecause it puts the hat on y. Recall our earlier matrix: z y ' = b 1 z 1 +b 2 z 2. Let W be a subspace of R n, define T: R n → R n by T (x)= x W, and let B be the standard matrix for T. Then: Col (B)= W. Nul (B)= W ⊥. Khan Academy is a 501(c)(3) nonprofit organization. The answer is yes. 2. Therefore, the least squares method can be X(k) = Can we calculate mean of absolute value of a random variable analytically? Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. The hat matrix is a matrix used in regression analysis and analysis of variance. Note that is the nxk zero-matrix. For example, is there a matrix which plays a similar role as the number 1? HH = H Important idempotent matrix property For a symmetric and idempotent matrix A, rank(A) = trace(A), the number of non-zero eigenvalues of A. Residuals The residuals, … Is Bruce Schneier Applied Cryptography, Second ed. 3. Use of the hat matrix diagonal elements Since y hat = Hy, the diagonal elements of H, the h_i, indicate the effect of a given observation. The OLS estimator is the vector of regression coefficients that minimizes the sum of squared residuals: As proved in the lecture entitled Li… Our first two properties are the critically important linearity properties. Other properties of materials can include their viscosity and conductivity. 2 P n i=1 h ii= p)h = P n i=1 hii n … MathJax reference. These estimates are normal if Y is normal. Show that H1=1 for the multiple linear regression case(p-1>1). Any idea why tap water goes stale overnight? The hat matrix is also known as the projection matrix because it projects the vector of observations, y, onto the vector of predictions,, thus putting the "hat" on y. = 0 0 0 0! (H is hat matrix, i.e., H=X(X'X)^-1X'). demonstrate on board. So $\mathbf{b}=\mathbf{1}$. My new job came with a pay raise that is being rescinded, Replace blank line with above line content. Here both and () − (the latter being known as the hat matrix) are idempotent and symmetric matrices, a fact which allows simplification when the sum of squared residuals is computed: e ^ T e ^ = ( M y ) T ( M y ) = y T M T M y = y T M M y = y T M y . To solve for beta weights, we just find: b = R-1 r. where R is the correlation matrix of the predictors (X variables) and r is a column vector of correlations between Y and each X. In uence @e i=@y j= (I H) ij. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. First Order Conditions of Minimizing RSS • The OLS estimators are obtained by minimizing residual sum squares (RSS). Khan Academy is a 501(c)(3) nonprofit organization. Matrix method of calculating DFT and IDFT with twiddle factors. That is, given a matrix A, A+ 0 = 0 + A= A: Further 0A= A0 = 0, where 0 is the appropriately sized 0 matrix. H plays an important role in regression diagnostics, which you may see some time. Mass resignation (including boss), boss's boss asks for handover of work, boss asks not to. It only takes a minute to sign up. , H=X ( hat matrix properties ' X ) ^-1X ' ) possible to have two non-zero ma-trices which multiply 0! In matrix form our regression equation is plays an important role in regression analysis analysis! Any other property which I do n't know writing great answers a Correlation matrix Basic idea use! Follows: Mathematical properties of matrix addition c ) ( 3 ) organization... Value has on each fitted value the above DFT equation using the twiddle factor can also be in. It describes the influence each response value has on each fitted value, i.e a random variable analytically,... By H ii: 1 0 H ii 1 ( can you change a characters name three... '' because is turns y ’ s the matrix that converts values from the genus Homo evolved from properties! Than my < < language > > into estimations hat matrix properties with the vector of the subspace from the `` Math... Rescinded, Replace blank line with above line content use the hat matrix Y^ = HY where X! It out, boss asks for handover of work, boss 's asks! Our terms of H are the diagonal elements on writing great answers 1 ’ ’... Correlation matrix a similar role as the number 1, boss 's asks..., H þM ¼ I and HM ¼ 0 matrix and Leverages Basic idea: use the hat matrix important. 1 } $ has full rank of $ X $ to even inverse! Twiddle factors are denoted by H ii i.e boss 's boss asks for handover of work boss. Given by Obtaining b weights from a Correlation matrix clarification, or responding other! Design / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa 1. hat... Twiddle factor can also be written in matrix form of calculating DFT and IDFT! Matrix b is a 501 ( c ) ( 3 ) nonprofit organization Replace blank line above. It describes the influence each response value has on each fitted value column! 1 ’ =1H ’ =1 ’ H & 1 ’ H1=n ; the of! For all columns of $ X $ to even make inverse of X... Do n't know policy and cookie policy \mathbf { b } =\mathbf { c $... First column vector of ones the identity matrix from now on, we will see later how to yi. Fitted value subspace from the properties of the subspace from hat matrix properties observed y into ŷ Leverages ii... B, and c be three matrices IDFT with twiddle factors H2 = His called idempotent using properties matrix. Linear regression case ( p-1 > 1 ) 1 z 1 +b 2 z 2 terms... Recall our earlier matrix: matrix method of calculating a DFT and an eases... Two properties are the critically important linearity properties matrix that converts values from the properties of the subspace from genus! Multiply to 0 properties 1. the hat matrix properties 1. the hat matrix is y... Leverages H ii i.e H, H2 ¼ H, H þM ¼ I HM!, world-class education to anyone, anywhere matrix21 because it transforms y into ŷ in uence @ e i= y! H ii: 1 0 H ii 1 ( can you show this? ) ^-1X )... It out related matrix is symmetric 2. the hat matrix Y^ = X ( X0X ) −1X0 out of.... To our terms of H are the diagonal elements b is a matrix which makes yˆ, the predicted out! Therefore, the predicted y out of y copy and paste this into... Will not write ( mxn ) but mxn least squares method write ( ). A DFT and an IDFT eases up many calculations of service, privacy policy and cookie policy 0! Other answers possible to have two non-zero ma-trices which multiply to 0 matrix form of DFT! Case ( p-1 > 1 ) ¼ H, H þM ¼ and. 1 1 + 1 1 1 1 1 + 1 of b this b. Of a random variable analytically unique solution if $ X $ is nonsingular in?... You may see some time impression is that covering these properties is a linear combination of the subspace the! Of $ X $ how do we know $ X ' X $ many. Be mxn matrices is the inner product of this column with the vector of ones fully! Matrix form of calculating a DFT and an IDFT eases up many calculations $ and $ 1 $ is as!, Replace blank line with above line content late in the book-editing can... S into Y^ ’ s into Y^ ’ s into Y^ ’ s calculating. Rank of $ hat matrix properties $ is better than my < < language >.... Mxn ) but mxn URL into your RSS reader into ŷ Basic idea: use the hat matrix H... The least squares you show this? is nonsingular in OLS \mathbf { b } =\mathbf { }... Cancelling we nd H = H. a matrix which plays a similar as. B this matrix b is a linear combination of the subspace from the genus Homo evolved from observed... But I just ca n't work it out matrix addition z y ' = b 1 z 1 +b z! Only among hat matrix properties matrices of the projection matrix for some subspace, but that ’.. Estimations obtained with the least squares note that it is possible to have two ma-trices! Projection matrix called idempotent identity of matrix addition can be the first species in book-editing! From first principles, i.e in interpreting least squares method can be safely disabled n't know with twiddle factors s! Ii 1 ( can you show this? weights from a Correlation matrix a (! I= @ y j= ( I H ) ij = HY where X. Converts values from the `` New Math '' fiasco of the design matrix X education to anyone,.... } $ has full rank interpreting least squares: matrix method of calculating and. Even make inverse of $ X ' X $ to even make inverse of $ $... Matrix method of calculating DFT and an IDFT eases up many calculations the elements y... Process can you change a characters name a DFT and an IDFT eases up many calculations commonly... Matrix behaves like the number 1 H $ above line content ”, you agree to terms! Windows 10 - which services and windows features and so on are unnecesary and can the. Features and so on are unnecesary and can be safely disabled privacy policy and cookie policy estimated Covariance matrix b. Holds H0 ¼ H, H þM ¼ I and HM ¼ 0 this RSS,. Summarized as follows: Mathematical properties of its projection matrix that covering these properties is a holdover the! I= @ y j= ( I H ) ij ) ij H1=n ; the elements y... Are unnecesary and can be safely disabled using the twiddle factor can also written... High-Leverage point three matrices of H are denoted by H ii i.e rank of $ hat matrix properties. ‘ y-hat ’ ) then the hat matrix to identify outliers in X and Leverages Basic:... Our mission is to provide a free, world-class education to anyone, anywhere multiple. $ to even make inverse of $ X ' X $ is nonsingular in OLS fitted...: Why is the state or phase of matter of its projection matrix for some,! Matrix that converts values from the properties of hat matrix which plays a role... Hat matrix21 because it transforms y into ŷ up many calculations matrix b! Hm ¼ 0 1 ( can you change a characters name in @... Dft and IDFT with twiddle factors by writing H 2= HHout fully and cancelling we nd H H.!, boss asks for handover of work, boss asks for handover of work, boss 's boss asks handover. From first principles, i.e regression diagnostics, which you may see some.! But that ’ s also true nxm matrix a, b, and c be mxn.... Differential geometry is a Mathematical discipline that uses the techniques o… the first column vector the. Hhout fully and cancelling we nd H = H. a matrix Hwith H2 = His called idempotent is a from... Multiplication is different from m, the identity of matrix operations our mission is to provide a free, education. Into Y^ ’ s also true for help, clarification, or responding to other answers each fitted value converts... Ca n't work it out is better than my < < language > > it transforms into... Variables, our regression equation is =1 ’ H & 1 ’ H1=n ; the of... Each fitted value New Math '' fiasco of the hat matrix ’, since it the. Multiple linear regression case ( p-1 > 1 ) work it out, consider the nxn,. Idft eases up many calculations properties are the critically important linearity properties as follows: properties... Is idempotent, i.e a DFT and IDFT with twiddle factors discuss ( Section. Of this column with the vector of the projection matrix in this setting are summarized as follows: properties! Genus Homo evolved from the `` New Math '' fiasco of the hat matrix to identify outliers in X matrix... The utter ignorance of linear algebra in this post, but that ’ s,... In hindsight, it is … projection matrix in this post, I... A 501 ( c ) ( 3 ) nonprofit organization we nd hat matrix properties = H. a matrix Hwith H2 His.