3 ways to characterize the matrix of relative to bases
| and |
The transport diagram: where
| -th coordinate of relative to |
Let
Characteristic polynomial:
Eigenvalues — roots of : , ,
Eigenvectors (in order):
Begin with the diagonal matrix having and as eigenvalues.
and are the eigenvectors of .
Perform a change of basis where the columns of form the angle .
Note:
Recall for any matrices and one has where as a subscript on a matrix indicates the -th column.
The -th column of a diagonal matrix is a scalar times the -th column of the identity matrix.
Apply these observations to the relation to understand why the columns of are eigenvectors of .
Characteristic polynomial:
Only one eigenvalue:
Eigenspace for the eigenvalue has dimension
No basis consisting of eigenvectors
Hence, not diagonalizable.
Find an orthogonal matrix and a diagonal matrix such that
What is the largest value achieved on the unit sphere by the function
Characteristic polynomial:
Eigenvalues — roots of : , ,
Eigenvectors (in order):
Since is an orthogonal matrix, is on the unit sphere if and only if is on the unit sphere.
Write x = Uv and compute as a function of .
Since is an orthogonal matrix, .
The maximum value of when is .