Misconceptions about diagonal and tridiagonal matrices


A reader wrote: “I purchased on lulu the 2nd edition of your Introduction to Matrix Algebra for self study, and the book just arrived. I started reading it and found some annoying errors. For example on Chapter 1, page 5: for the first (diagonal) matrix, why is there a zero located in a33, when you defined on the previous page that only diagonal entries of square matrix can be non-zero (this answer is different on your free online pdf) . Also, on page 6 for the tridiagonal matrix, why is there a zero located in the diagonal below the major diagonal? I was wondering if you can provide me with the list of errors and corrections, because it’s going to be very difficult to study the material on my own and the errors in the book just makes it more frustrating.”

My answer: “There is no erratum issued yet on the book.

A diagonal matrix is diagonal based on the nondiagonal elements being zero. The diagonal elements have no restrictions. They can be zero or nonzero.

A tridiagonal matrix is a square matrix in which all elements not on the following are zero – the major diagonal, the above the major diagonal, and the diagonal below the major diagonal. The major diagonal, the diagonal above the major diagonal, and the diagonal below the major diagonal have no restrictions. They can be zero or nonzero.

The concerns you have raised are some of the common misconceptions students develop about these special matrices.

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Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.

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