The reason the matrix in Exercise 4.6 is diagonal is that the new basis vectors are all eigenvectors. Recall the definition:
Given a linear transformation from a vector space to itself, an eigenvector is a non-zero vector such that for some scalar , called an eigenvalue.
It is understood that eigenvectors of a square matrix refer to the eigenvectors of the associated linear transformation , defined by , using the standard basis to write vectors in .
In MATH105, techniques were developed to find all eigenvalues and eigenvectors of real square matrices, first by solving the polynomial equation (for ), and then for each eigenvalue, finding all eigenvectors by solving a system of linear equations in the coefficients; these techniques still work over arbitrary fields . Recall that is called the characteristic polynomial of . It is a degree polynomial with coefficients in . One of the main benefits of finding eigenvectors is the following:
If a vector space has a basis consisting of eigenvectors of some linear transformation , then
where is the eigenvalue of .
Since , the th column of the matrix is
, where the is in the th position. So all of the ’s are along the diagonal of , with zeros elsewhere. ∎
Find the eigenvalues, and their corresponding eigenvectors (known as an eigenspace), for each of the following matrices.
For each of the matrices in Exercise 4.12, decide whether or not has a basis consisting of eigenvectors.
Let be the vector space of matrices over a field . Let be the transpose, defined by . Then is a linear transformation. Can you find a basis of in which is diagonal?
[End of Exercise]