Mathematical Tools for Physics - Department of Physics - University

7—Operators and Matrices 154 the other. This is no different from the way you look at ordinary real valued functions. The expone ...

7—Operators and Matrices 155 Here the notationR~θrepresents the function prescribing a rotation byθabout the axis pointing along ...

7—Operators and Matrices 156 What is the area of this parallelogram? I’ll ask a more general question. (It isn’t really, but it ...

7—Operators and Matrices 157 do it with your left hand. You cannot move one of these and put it on top of the other (unless you ...

7—Operators and Matrices 158 Theorem: Up to an overall constant factor, this function is unique. An important result is that the ...

7—Operators and Matrices 159 and the last term is zero. Translate this into components. Use the common notation for a determinan ...

7—Operators and Matrices 160 an operatorf, then consider the determinant ofM=I+f, whereIis the identity. This combination is ve ...

7—Operators and Matrices 161 The components ofM in this basis are ( 5 − 1 − 2 0 ) It doesn’t look at all the same, but it repres ...

7—Operators and Matrices 162 To be diagonal simply means thatfki= 0for alli 6 =k, and that in turn means that all but one term i ...

7—Operators and Matrices 163 Here I arbitrarily wrote the equation for three dimensions. That will change with the problem. Put ...

7—Operators and Matrices 164 Example: Coupled Oscillators Another example drawn from physics: Two masses are connected to a set ...

7—Operators and Matrices 165 7.10 Change of Basis In many problems in physics and mathematics, the correct choice of basis can e ...

7—Operators and Matrices 166 The final equation comes from the preceding line. The coefficients of~ekmust agree on the two sides ...

7—Operators and Matrices 167 This is the eigenvector equation that you’ve supposedly already solved. The first column of the com ...

7—Operators and Matrices 168 This provides only one eigenvector, a multiple of ( 1 0 ) . You need two for a basis. Change this m ...

7—Operators and Matrices 169 with familiar roots α= ( −b± √ b^2 − 4 km ) / 2 m If the two roots are equal youmaynot have distinc ...

7—Operators and Matrices 170 But this assumes that you already know the ratings of the sites, and that’s what you’re trying to f ...

7—Operators and Matrices 171 chapter on Fourier series, section5.3, but it didn’t appear under this name. You will become famili ...

7—Operators and Matrices 172 Problems 7.1 Draw a picture of the effect of these linear transformations on the unit square with v ...

7—Operators and Matrices 173 and repeat the problem, finding components of the first and second derivative operators. Verify an ...