Question

Find the eigenvalues and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer. $$\left(\begin{array}{rr}2& 2\\ 2& -1\end{array}\right)$$

Short answer

Eigenvalues are There are two possible values: to some actual sentence then accurately in between

Step-by-step solution

Define the matrix

Start by identifying the given matrix, which is

Key concepts

Matrix Eigenvalues

Matrix eigenvalues are special numbers associated with a square matrix. To find an eigenvalue of a matrix, we need to solve the characteristic equation, which is determined by the matrix itself. For a matrix $A$, the characteristic equation is given by: $$\text{det}(A-\lambda I)=0$$, where $\lambda$ represents the eigenvalue and $I$ is the identity matrix of the same dimension as $A$. In our exercise's provided matrix: $$A=\left(\begin{array}{ccc}2& 22& -1\end{array}\right),$$ we use this formula to find the eigenvalues of $A$. By setting up the determinant, you will find the necessary solutions of the equation. Solving this will reveal the eigenvalues needed to understand the behavior of the matrix.

Matrix Eigenvectors

Matrix eigenvectors correspond to eigenvalues and are vectors that, when multiplied by the matrix, result in a vector that is a scalar multiple of the original vector. This scalar multiple is the eigenvalue. Given an eigenvalue $\lambda$ of a matrix $A$, the corresponding eigenvectors $v$ can be found by solving the equation: $$(A-\lambda I)v=0$$. Using our earlier example $$A=\left(\begin{array}{ccc}2& 22& -1\end{array}\right)$$, we substitute each eigenvalue back into the equation to solve for $v$. Eigenvectors help in expressing the matrix in a simpler form, especially useful in various computations like diagonalization.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations. It plays a crucial role in understanding eigenvalues and eigenvectors. This area of study involves matrices, their properties, and the relationship between sets of linear equations. In our matrix eigenvalue problem, linear algebra provides the tools to find eigenvalues and eigenvectors, which give insights into the transformation properties of the matrix. The decomposition of matrices using eigenvalues and eigenvectors can simplify complex problems and is foundational for many applications in engineering, physics, and computer science.

Matrix Eigenvalue Problems

Matrix eigenvalue problems involve finding the eigenvalues and eigenvectors of a given matrix. These problems are essential because they can reveal important properties about the matrix's structure and the transformations it represents. In our exercise, we need to solve the characteristic equation to find the eigenvalues. Once found, these eigenvalues can be used to determine the corresponding eigenvectors by solving a system of linear equations. Understanding these problems is key to simplifying and solving many practical issues in different fields, including data analysis, mechanics, and quantum physics.