Question

Let \(A\) be a real \((2 \times 2)\) matrix having repeated eigenvalue \(\lambda_{1}=\lambda_{2}=\alpha\) and a full set of eigenvectors, \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). Show that \(A=\alpha I\). [Hint: Let \(T=\left[\mathbf{x}_{1}, \mathbf{x}_{2}\right]\) be the invertible ( \(2 \times 2\) ) matrix whose columns are the eigenvectors. Show that \(A T=\alpha T\).]

Short answer

Question: Prove that a 2x2 matrix A with a repeated eigenvalue and a full set of eigenvectors must equal αI, where I is the identity matrix. Answer: To show that A = αI for a 2x2 matrix A with repeated eigenvalue α and a full set of eigenvectors, we can define an invertible matrix T with columns equal to the eigenvectors and show that AT = αT. After multiplying by the inverse of T and simplifying, we can observe that AI = αI, which implies that A = αI.

Step-by-step solution

Define the given matrix and its eigenvectors

We have a real 2x2 matrix \(A\) with eigenvalue \(\lambda_1 = \lambda_2 = \alpha\) and eigenvectors \(\mathbf{x}_1\) and \(\mathbf{x}_2\). The definition of an eigenvector is: \(A\mathbf{x}_1=\lambda_1\mathbf{x}_1=\alpha\mathbf{x}_1\) \(A\mathbf{x}_2=\lambda_2\mathbf{x}_2=\alpha\mathbf{x}_2\)

Define the invertible matrix T and its inverse

Create a matrix \(T=\begin{bmatrix}\mathbf{x}_{1} & \mathbf{x}_{2}\end{bmatrix}\), which has eigenvectors as columns. Since \(\mathbf{x}_1\) and \(\mathbf{x}_2\) are a full set of eigenvectors, T is invertible. Let \(T^{-1}\) denote the inverse of T.

Multiply matrix A by T

Now, multiply matrix A by matrix T: \(AT=\begin{bmatrix}A\mathbf{x}_{1} & A\mathbf{x}_{2}\end{bmatrix}=\begin{bmatrix}\alpha\mathbf{x}_{1} & \alpha\mathbf{x}_{2}\end{bmatrix}=\alpha\begin{bmatrix}\mathbf{x}_{1} & \mathbf{x}_{2}\end{bmatrix}=\alpha T\)

Show that A = αI

We have shown that \(AT = \alpha T\). Now, multiply the equation by the inverse of T from the right side: \(A(TT^{-1})=\alpha T(T^{-1})\) Since \(TT^{-1}=I\), we get: \(AI = \alpha I\) Now, we can see that A = \(\alpha I\). This proves that a 2x2 matrix A with a repeated eigenvalue and a full set of eigenvectors must equal \(\alpha I\).

Key concepts

Matrix Diagonalization

Matrix diagonalization is a process that simplifies matrices by converting them into a diagonal form. This is achieved when a matrix is expressed as a product of three matrices: an invertible matrix containing eigenvectors, a diagonal matrix containing eigenvalues, and the inverse of the first matrix. Diagonal matrices are much simpler to work with because most operations, like raising to powers, are straightforward.

In the context of our exercise, we were given a matrix with repeated eigenvalues and a full set of eigenvectors. The objective was to show that the matrix could be expressed as a scaled identity matrix. By constructing the matrix \( T \) from the eigenvectors, we demonstrated that the original matrix \( A \) simplifies to \( \alpha I \) upon diagonalization. This aligns perfectly with the power of diagonalization: simplifying and solving matrix equations effectively.

Matrix diagonalization serves as a powerful tool in many applications such as solving systems of differential equations, computing matrix functions, and simplifying matrix operations.

Linear Algebra

Linear algebra is a field of mathematics dedicated to the study of vectors, vector spaces, and linear transformations. It provides fundamental concepts and structures that are critical in understanding matrix operations. In this case, our use of eigenvalues and eigenvectors relies heavily on linear algebra principles.

One core concept of linear algebra is the eigenvalue-eigenvector relationship, which helps in understanding how linear transformations affect certain vectors. When we say that a vector \( \mathbf{x} \) is an eigenvector of a matrix \( A \), it means that transformation by \( A \) scales \( \mathbf{x} \) by a factor known as an eigenvalue, \( \lambda \). This underpins many processes in diagonalization, as explored in the exercise.

Linear algebra also introduces the idea of vector spaces and bases, which explains why a complete set of linearly independent eigenvectors makes the matrix \( T \) invertible. When these concepts are applied, they form the backbone of powerful matrix-based methods fundamental to science and engineering.

Invertible Matrix

An invertible matrix, also referred to as a non-singular or non-degenerate matrix, is one that has an inverse. The inverse of a matrix \( A \) is a matrix \( B \) such that when \( A \) is multiplied by \( B \), the resulting product is the identity matrix \( I \). Not all matrices are invertible; a matrix must be square and have full rank to qualify.

In our step-by-step solution, we used an invertible matrix \( T \) built from the eigenvectors of \( A \). The invertibility of \( T \) was crucial because it allowed us to transform and then reverse-transform matrices without losing information. This transformation was a pivotal step in showing that \( A \) could be simplified to \( \alpha I \).

Matrices that are invertible hold significant importance across many mathematical and computational applications. They ensure the solvability of systems of linear equations, and understanding them enriches one's ability to manipulate and utilize linear transformations effectively.