Is idempotent matrix always diagonalizable?

An idempotent matrix satisfies the equation It has two distinct roots 0 & 1. This is minimal polynomial of A, except when A is either zero or identity matrix, both of which are diagonalizable as they are diagonal matrices. Hence, any idempotent matrix is diagonalizable.

Is every invertible matrix diagonalizable?

Invertibility does not imply diagonalizability: Any invertible matrix with Jordan blocks of size greater than will fail to be diagonalizable. So the minimal example is any with . Diagonalizability does not imply invertibility: Any diagonal matrix with a somewhere on the main diagonal is an example.

Is a diagonal matrix always diagonalizable?

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. That is, A is diagonalizable if there is an invertible matrix P and a diagonal matrix D such that. A=PDP^{-1}.

Why is an invertible matrix not diagonalizable?

It has two linearly independent columns, and is thus invertible. At the same time, it has only one eigenvector: v=[10]. Since it doesn’t have two linearly independent eigenvectors, it is not diagonalizable.

How do you prove idempotent?

A matrix A is idempotent if and only if all its eigenvalues are either 0 or 1. The number of eigenvalues equal to 1 is then tr(A). Since v = 0 we find λ − λ2 = λ(1 − λ) = 0 so either λ = 0 or λ = 1. Since all the diagonal entries in Λ are 0 or 1 we are done the proof.

What is the condition for a matrix to be diagonalizable?

A linear map T: V → V with n = dim(V) is diagonalizable if it has n distinct eigenvalues, i.e. if its characteristic polynomial has n distinct roots in F. of F, then A is diagonalizable.

What are the conditions for a matrix to be diagonalizable?

A is diagonalizable if and only if its minimal polynomial can be factored into a product of linear factors (without passing to an extension field), with no repeated roots. In particular, for real matrices, that means its minimal polynomial has only real roots, and no repeated roots.