Question

A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 1 & -2 \\ -2 & 4 \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right] \end{array} $$

Short answer

The eigenvalue \(\lambda\) for the eigenvector \(\vec{x}\) is 0.

Step-by-step solution

Understand the Eigenvalue Problem

The eigenvalue problem is expressed as \(A\vec{x} = \lambda \vec{x}\). Here, \(A\) is the matrix, \(\vec{x}\) is the eigenvector, and \(\lambda\) is the eigenvalue that we want to find. The goal is to determine \(\lambda\) for the given \(\vec{x}\).

Multiply the Matrix by the Eigenvector

Calculate the product \(A\vec{x}\). \[ A\vec{x} = \begin{bmatrix} 1 & -2 \ -2 & 4 \end{bmatrix} \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 1 \times 2 + (-2) \times 1 \ -2 \times 2 + 4 \times 1 \end{bmatrix} = \begin{bmatrix} 2 - 2 \ -4 + 4 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \]Thus, \(A\vec{x} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\).

Relate the Product to the Eigenvalue Equation

According to the eigenvalue equation \(A\vec{x} = \lambda \vec{x}\), since \(A\vec{x} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\) and \(\vec{x} = \begin{bmatrix} 2 \ 1 \end{bmatrix}\), it follows that: \[ \lambda \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \]This implies the equation zeroes out each component, i.e., \(2\lambda = 0\) and \(\lambda = 0\).

Solve for the Eigenvalue

Since the equation is \(2\lambda = 0\), we solve for \(\lambda\) by dividing both sides by 2. This results in:\[ \lambda = 0 \]Thus, the eigenvalue \(\lambda\) corresponding to the given eigenvector \(\vec{x}\) is zero.

Key concepts

Eigenvectors

Eigenvectors are special vectors that remain proportional to themselves after a linear transformation is applied. They are linked to square matrices and form the backbone of one important field: Eigenvalue problems in linear algebra. When you multiply a matrix by its eigenvector, the direction of the eigenvector is unchanged, though its magnitude can be altered.

• **Definition**: If matrix \(A\) transforms vector \(\vec{x}\) such that \(A\vec{x} = \lambda \vec{x}\), where \(\lambda\) is a scalar, then \(\vec{x}\) is an eigenvector of \(A\).
• **Significance**: Eigenvectors are pivotal in numerous scientific and engineering applications. Whether it involves stability analysis, vibrations, facial recognition, or page ranking, understanding eigenvectors helps bridge practical problems to their mathematical solutions.
• **Computational Note**: Not every matrix has eigenvectors, and some have complex eigenvectors depending on their characteristics. However, symmetric matrices always have real eigenvectors.

Matrix Multiplication

Matrix multiplication is a unique operation that combines two matrices to create a new matrix. Unlike basic multiplication, it requires specific rules.

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second.

• **Steps**: Position each element from the rows of the first matrix and columns of the second matrix. Multiply these elements directly, and then sum all product results to produce each entry of the resulting matrix.

For example, when you compute \(A\vec{x}\) with matrix \(A\) and vector \(\vec{x}\), it involves:

• Calculating the dot product of rows of \(A\) with \(\vec{x}\).
• Arranging them in a new vector or matrix form.

This task helps you understand how transformation applies through linear algebra systems, enabling the derivation of eigenvectors and eigenvalues effectively.

Linear Algebra

Linear algebra is the study of vectors, vector spaces, and linear transformations. It provides a framework for understanding two core concepts mentioned before: eigenvectors and matrix multiplication.

• **Vectors and Matrices**: Vectors are fundamental elements representing positions, directions in space, or various quantities. Matrices, meanwhile, are arrays of numbers or expressions that can represent linear transformations or systems of equations.
• **Linear Transformations**: These are functions that map vectors to vectors in a way that preserves the operations of vector addition and scalar multiplication. They are represented through matrices, and help in tasks like rotating or scaling geometrical shapes.
• **Core Importance**: Linear algebra is crucial in modern analysis, from solving equations with multiple variables to simulating natural processes. Many phenomena across physics, computer graphics, and machine learning are based on principles derived from this branch of mathematics.