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Description
This programme looks at how matrix multiplication can be represented by a geometric transformation and how, by decomposition, the calculation of powers of matrices can be simplified.
Module code and title: M101, Mathematics: a foundation course M101; 19; 1985 14-07-1985 1985 00:24:23 + Show more... Andrew Barker Bob Coates; Mike Crampin BBC Open University Eigenline; Eigenvalue; Matrices; Models; Powers; Scaling It shows how the calculation of powers of certain 2x2 matrices can be simplified by making use of the QDQ-1 representation and the geometrical interpretation. The programme begins be reminding the student of the properties of a 2 x 2 matrix which was investigated in the pre-programme work. The matrix has two skew eigenlines - lines which under the matrix transformation remain in the same position. It then takes a different matrix with eigenlines at right-angles and considers what happens when points not on the eigenlines are transformed. The matrix can be expressed as the product of three matrices, representing a rotation, a scaling along the axes, and a rotation. This is illustrated by a studio model which shows how points on the eigenlines are transformed. An animation then shows how the three successive transformations do represent the original matrix transformation and how powers of this matrix can be obtained through a series of geometrical transformations. The presenter then considers the matrix which stated the programme. In an analogous way to the previous demonstration an animation shows how points not on the eigenlines are transformed. The presenter then sets up the QDQ-1 representation of the matrix where Q is a transformation which takes the axes to the eigenlines and D is a scaling along the axes and an animation shows how this can be used to calculate the square and cube of the matrix. Finally the nth power is represented as QDQ-1. HOU5024 FOUM221D 2477 no