Chapter 002, Linear Equations and Eigensystems

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In this simulation study, we consider nearly identical conditions to the first, with the exception that the item discrimination parameters, a i , are large in size, indicating excellent item discrimination. As in the previous study, we considered six conditions comprised of: 4, 6, 8, 16, 32, and 64 test items. As before, for each condition, we have simulated responses to the test items. For each condition, we computed the covariance matrix of the simulated responses and numerically calculated the eigenvalues of the generated covariance matrix for each condition.

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It is interesting to note that there is a much closer correspondence between the two sets of eigenvalues under these conditions. Further, this relationship becomes stronger as the number of equally spaced items increases, yielding nearly a perfect match to the maximal eigenvalue as the number of items reaches 32 and This study illustrates that our analytic results, which are derived under the strong assumption of uniformly distributed response patterns, may be useful as an approximation even when the ability parameter is normally distributed. This approximation is best when the difficulty range of the items are within a single standard deviation of the mean and the items have excellent discriminability.


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Our Matlab code for generating these graphs and exploring other configurations is available as an online supplement. We derived closed-form solutions for the eigenvalues and eigenvectors of the covariance matrix of dichotomous Guttman items, under a uniform sampling assumption. We demonstrated that these eigenvalues and eigenvectors are simple trigonometric functions of the number of items, n.

Our results parallel those of Zwick , who examined the eigenvalues of the correlation matrix of dichotomous Guttman items under the same uniformity assumptions.

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It remains an open question whether the eigenvectors of the correlation matrix, as investigated by Zwick , can also be solved for explicitly. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Bulletin ExploreCourses.


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  • CME Linear Algebra and Partial Differential Equations for Engineers ENGR B Linear algebra: matrix operations, systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, eigensystem analysis, normal modes. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations.

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    Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications from various engineering fields. Instructors: Khayms, V. PI ; Chen, E. TA ; Fournier-Bidoz, E. TA Additional Resources: Login to view additional resources. Page: 1. Browse by subject. Free download. Book file PDF easily for everyone and every device.

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      Ortega, W. ACM Symp. International Symp. Pan, D. Ivolgin, B. Murphy, R. Rosholt, I. Taj-Eddin, Y. Tang, X. Pan, B. Rosholt, Y. Yan, W. Pan in Proc. Pan, J. Scavo, J. Monthly , , —, ACM , 11 , 12—14, Personalised recommendations. Cite paper How to cite?