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Matrix Di erentiation

Matrix Differentiation( and some other stuff )Randal J. BarnesDepartment of Civil Engineering, University of MinnesotaMinneapolis, Minnesota, USA1 IntroductionThroughout this presentation I have chosen to use asymbolic Matrix notation. This choicewas not made lightly. I am a strong advocate of index notation, when appropriate. Forexample, index notation greatly simplifies the presentation and manipulation of differentialgeometry. As a rule-of-thumb, if your work is going to primarily involve differentiationwith respect to the spatial coordinates, then index notation is almost surely the the present case, however, I will be manipulating large systems of equations in whichthe Matrix calculus is relatively simply while the Matrix algebra and Matrix arithmetic ismessy and more involved.

CE 8361 Spring 2006 Proposition 4 Let A be a square, nonsingular matrix of order m. Partition A as A = " A 11 A 12 A 21 A 22 # (20) so that A 11 is a nonsingular matrix of order m 1, A 22 is a nonsingular matrix of order m 2, and m 1 +m 2 = m. Then

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