Question

If \(A\) is diagonalizable and 0 and 1 are the only eigenvalues, show that \(A^{2}=A\).

Short answer

Since all eigenvalues are 0 or 1, \(A^2 = A\).

Step-by-step solution

Understanding the Given Information

The problem states that matrix \(A\) is diagonalizable and its eigenvalues are 0 and 1. This means that \(A\) can be expressed in the form \(A = PDP^{-1}\), where \(D\) is a diagonal matrix containing the eigenvalues of \(A\), and \(P\) is an invertible matrix. The diagonal matrix \(D\) will have only 0s and 1s on its diagonal since those are the only eigenvalues.

Formulate the Diagonal Matrix D

Since \(D\) is a diagonal matrix with eigenvalues 0 and 1, it can be represented as \(D = \begin{pmatrix} \lambda_1 & 0 & 0 \ 0 & \lambda_2 & 0 \ 0 & 0 & \lambda_3 \end{pmatrix}\) where each \(\lambda_i\) is either 0 or 1. Every diagonal entry in \(D\) is an eigenvalue of \(A\).

Calculate \(A^2\) Using the Diagonal Form

Since \(A = PDP^{-1}\), we can compute \(A^2\) as follows:\(A^2 = (PDP^{-1})(PDP^{-1}) = PD(P^{-1}P)DP^{-1} = PD^2P^{-1}\).The matrix \(D^2\) is just the matrix \(D\) squared. But because all diagonal entries are 0s and 1s, squaring them leaves the matrix unchanged. Thus, \(D^2 = D\).

Express \(A^2 = A\)

From the previous step, we found that \(A^2 = PD^2P^{-1} = PDP^{-1} = A\). Therefore, we conclude that \(A^2 = A\).

Key concepts

Eigenvalues

Eigenvalues are a fundamental concept in matrix algebra, representing special scalars associated with a matrix. When you multiply a matrix by a vector, the vector may change direction and length. However, in some cases, the direction does not change; instead, the vector is merely scaled by a number, which is called an eigenvalue. This occurs when:

• \( Av = \lambda v \)

where \( A \) is a matrix, \( v \) is the eigenvector, and \( \lambda \) is the eigenvalue. A matrix can have several eigenvalues and corresponding eigenvectors.
In our exercise, matrix \( A \) is described as having eigenvalues of 0 and 1. This tells us two things:

• If an eigenvalue is 0, then any associated eigenvector is within the null space of the matrix.
• If an eigenvalue is 1, any associated eigenvector remains unchanged by \( A \).

These properties are crucial in proving additional matrix behaviors, as they dictate how the matrix transforms vectors and its own powers.

Matrix Diagonalization

Matrix diagonalization is a method of transforming a given matrix into a diagonal form, which is easier for calculations, especially for powers of matrices. A matrix is said to be diagonalizable if it can be expressed in the form:

• \( A = PDP^{-1} \)

where \( D \) is a diagonal matrix with eigenvalues of \( A \) on its diagonal, and \( P \) is an invertible matrix composed of the eigenvectors of \( A \).
In our example, since matrix \( A \) is diagonalizable with eigenvalues 0 and 1, the matrix \( D \) would simply have these values on its diagonal. The beauty of diagonalization is that it simplifies exponentially complex operations. For power calculations, like squares and higher powers:

• \( A^2 = PDP^{-1}PDP^{-1} = PD^2P^{-1} \)

Since \( D^2 = D \) for diagonal entries that are either 0 or 1 (as squaring these values have no effect), it simplifies to:

• \( PD^2P^{-1} = PDP^{-1} = A \)

showing that \( A^2 = A \). This property beautifully illustrates the simplification powers of diagonalization for matrices with these eigenvalues.

Matrix Algebra

Matrix algebra involves a set of rules and operations that apply to matrices, extending various algebraic operations into multi-dimensional contexts. It's an essential part of linear algebra and encompasses operations like addition, multiplication, and finding inverses.
In our focus is the operation of matrix multiplication, particularly when dealing with diagonalizable matrices. When we work with a matrix like \( A = PDP^{-1} \), understanding how these components interact is key:

• \( P \) and \( P^{-1} \) sandwich the diagonal matrix \( D \), enabling transformation back and forth between the original and diagonal forms.
• Matrix multiplication is associative, which allows for simplifications like \( A^2 = PD^2P^{-1} \), seen in our exercise above.

What's fascinating about matrix algebra is how it facilitates operations that would be cumbersome in a pure numerical form. For instance, computing the square of a matrix directly can be cumbersome without the diagonalization process. However, by deploying the PD-formulation, calculations become efficient and more intuitive. This efficiency underscores the utility of matrix algebra in solving complex problems, particularly for engineering, physics, and computer science applications, where such operations are routinely necessary.