Eigenvalues and eigenvectors
DOWNLOAD
FREE
|
Eigenvalues and Eigenvectors
Introduction: A matrix represents a transformation of the plane. Some matrices represent rotations of the plane; some matrices represent reflections of the plane, and some matrices represent shears. Another general class of matrices represent stretches of the x,y-plane. A simple example of this is the matrix [Equation goes here - download the original to see it.] This matrix stretches the x-axis 3 times and stretches the y-axis twice, but in the opposite direction. [Diagram goes here - download the original to see it.] What happens to the x and y-axes also determines what happens to every other vector lying in the plane. For instance, since [Equation goes here - download the original to see it.] The matrix A has the effect of scaling this vector by 3 times in the x-direction and by -2 times in the y-direction. [Equation goes here - download the original to see it.] In other words, if we know that a matrix has the effect of stretching vectors along the two coordinate axes, we are then able to describe the effect of the matrix on any vector lying in the plane. However, pretty obviously, the case of stretches along both or even either the x and y-axes is the exception, rather than the rule. We seek a more general theory that will apply to all matrices that represent some form of systematic stretching of the plane. To illustrate this, consider the matrix [Equation goes here - download the original to see it.] If we try out at random some vectors and see what this matrix does to them, we may be at a loss to explain its effect on the plane as a whole. For example, this matrix sends the unit vector in the x-direction, [Equation goes here - download the original to see it.] , and the unit vector in the y-direction, [Equation goes here - download the original to see it.] , but this information does not really help us to visualise the effect of the transformation on the plane as a whole. [Diagram goes here - download the original to see it.] After some trial and error, however, the student may discover that there are lines, like the x and y-axes described in the first example, along which the effect of the transformation is simply to stretch (and possibly reflect, since a reflection is a negative stretch). Fo example, one such line which is mapped to itself by the matrix A, lies in the direction [Equation goes here - download the original to see it.]. Operating by on this vector by means of the matrix A [Equation goes here - download the original to see it.] by obtain another vector which is simply a multiple of the first. So, in the direction represented by the vector [Equation goes here - download the original to see it.] the matrix A has the effect of stretching the plane. Another vector which has this property is [Equation goes here - download the original to see it.] Along this vector we obtain another scalar multiple of itself by operating with A [Equation goes here - download the original to see it.] Since any two vectors are sufficient to define the whole 2-dimensional plane - they form a basis for the plane (or more technically, they form a basis for [Equation goes here - download the original to see it.] ), the entire effect of the transformation represented by the matrix A, can be summarised by saying that A stretches a vector by a factor of 2 in the direction of the vector [Equation goes here - download the original to see it.] and by a factor of 4 in the direction of the vector [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Since we have uncovered by trial and error this simple geometric description of the effect of a whole class of matrices on the x,y-plane, we seek a systematic algebraic technique to discover those lines that are mapped to themselves by such matrices, and the scale factors by which they are mapped
|
Contents of
Eigenvalues and eigenvectors
1 Eigenvalues and Eigenvectors
2 Determining eigenvalues and eigenvectors by use of the characteristic equation
3 Eigenvalues, eigenvectors, matrices and linear transformations
4 Diagonalisation
5 Power matrices
|
