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Unformatted text preview: Singular matrices... does a matrix always have an inverse? No! Matrices need not have an inverse If a matrix does not have an inverse, it is not invertible , or it is called singular Consider our expression for M 1 , M 1 = 1 det M C T M is singular if det M = 0 Geometric interpretation: If we consider the rows (or columns) of M to be vectors (consider a 3 3 matrix, the determinant is the volume of a polyhedron described by the vectors. Then = det M = 0 corresponds to the case where the three vectors lie all in a plane. This is related to the question of linear dependence/independence that we will address later. Patrick K. Schelling Introduction to Theoretical Methods Rotation matrix in two dimensions Linear transformations are important applications of linear algebra Coordinate transformations are important, including rotations Start with a vector ~ r = x i + y j and rotate by angle After rotation, x x and y y , or ~ r ~ R We can find the rotation matrix R ( ) and the transformation x y = cos  sin sin cos x y What is the inverse of this matrix? Think of the simplest idea Patrick K. Schelling Introduction to Theoretical Methods Inverse of rotation matrix... rotate back by angle ! We could rotate back with R 1 ( ) = R ( ) R ( ) = cos  sin sin cos R 1 ( ) = R ( ) = cos sin  sin cos Try it! R 1 ( ) R ( ) = cos sin  sin cos cos  sin sin cos = 1 0 0 1 Patrick K. Schelling Introduction to Theoretical Methods Multiple rotations, and another (equivalent) picture It is easy to show that R ( ) R ( ) = R ( + ) (see Problem 25, section 6) Moreover, R ( ) R ( ) R ( ) = R ( + + ), etc. In another picture, we can imagine rotating the axes by an angle , and leaving vector ~ r = x i + y j fixed In this picture, x and y define point in new coordinate system, and are given by x y = cos sin  sin cos x y Both of these pictures are often used... In one rotating the vector, and the other rotating the coordinate system...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
 Staff

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