I know that for a lower triangular matrix, its eigenvalues are equal to its diagonal elements. And if we assume a unit lower triangular matrix, its eigenvalues are all equal to 1 (since it is a unit lower triangular matrix).
My question is what are the eigenvectors of a unit lower triangular matrix? there exists a specific property about its eigenvectors?
For example, if we consider a unit lower triangular matrix A = [1,0,0; 4,1,0; 12,24,1].
Using matlab, I am calculating its eigenvalues and eigenvectors using the command eig. For example [e l] = eig(A). I am always getting a matrix e where all its elements are equal to 0 but the last line is equal to 1.
So does a unit lower triangular matrix can be expressed as the product of EDE'where E is the eigenvector matrix and D is the eigenvalues matrix and ' designates the transpose operator? But I am pretty sure that the elements of its eigenvector matrix should not be equal to 0.. Any help will be very appreciated!