Question

Is a triangular matrix diagonalizable?​

Answer 1

A set of matrices is said to be simultaneously diagonalizable if there exists a single invertible matrix such that is a diagonal matrix for every in the set.

Determine whether the matrix A is diagonalizable. If it is diagonalizable, then diagonalize A. That is, find a nonsingular matrix S and a diagonal matrix D such that, If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal Let A and B be n×n matrices.

The diagonalization theorem states that an matrix is diagonalizable if and only if has linearly independent eigenvectors, i.e., if the matrix rank of the matrix formed by the eigenvectors is . The following table gives counts of diagonalizable matrices of various kinds where the elements of may be real or complex.

For triangle:

Let A be such a matrix, let L be a scalar:

det(A-LI)=(a[1,1]-L)(a[2,2]-L)...(a[n,n]-L).

Each a[i,i] is an eigenvalue and by distinctness has multiplicity 1 in the characteristic equation, so A is diagonalisable.