Question

Find the eigenvalues and eigenvectors of (a) \(A_{1}=\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2\end{array}\right]\) (b) \(A_{2}=\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]\)

Short answer

Short Answer: For matrix \(A_1\), there is one eigenvalue, \(\lambda = 2\), with the corresponding eigenvector \(x=\begin{bmatrix}1\\0\\0\end{bmatrix}\). For matrix \(A_2\), there is one eigenvalue, \(\lambda = 2\). The corresponding eigenvectors can be expressed as \(x= \alpha \begin{bmatrix}1\\0\\0\end{bmatrix} + \beta \begin{bmatrix}0\\0\\1\end{bmatrix}\), where α and β are any real numbers.

Step-by-step solution

Calculate the determinant of \(A_1 - \lambda I\)

\(A_1-\lambda I=\begin{bmatrix}2-\lambda & 1 & 0\\0 & 2-\lambda & 1\\0 & 0 & 2-\lambda\end{bmatrix}\) Now we find the determinant: \(|A_1-\lambda I|=(2-\lambda)^3=0\)

Solve for the eigenvalues

\((2-\lambda)^3 = 0\) We can now solve for the eigenvalue: \(\lambda = 2\)

Substitute the eigenvalue back into the equation and solve for the eigenvector

\((A_1-\lambda I)x = 0\) Substitute the eigenvalue \(\lambda=2\): \(\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}x=0\) Let \(x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\), we get: x₂ = 0 x₃ = 0 Since x₁ is a free variable, the eigenvector corresponding to the eigenvalue λ=2 is: \(x= \begin{bmatrix}1\\0\\0\end{bmatrix}\) (b) Finding eigenvalues and eigenvectors of \(A_2\):

Calculate the determinant of \(A_2 - \lambda I\)

\(A_2-\lambda I=\begin{bmatrix}2-\lambda & 1 & 0\\0 & 2-\lambda & 0\\0 & 0 & 2-\lambda\end{bmatrix}\) Now we find the determinant: \(|A_2-\lambda I|=(2-\lambda)^3=0\)

Solve for the eigenvalues

\((2-\lambda)^3 = 0\) We can now solve for the eigenvalue: \(\lambda = 2\)

Substitute the eigenvalue back into the equation and solve for the eigenvector

\((A_2-\lambda I)x = 0\) Substitute the eigenvalue \(\lambda=2\): \(\begin{bmatrix}0 & 1 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}x=0\) Let \(x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\), we get: x₂ = 0 Since x₁ and x₃ are free variables, the eigenvector corresponding to the eigenvalue λ=2 can be: \(x= \alpha \begin{bmatrix}1\\0\\0\end{bmatrix} + \beta \begin{bmatrix}0\\0\\1\end{bmatrix}\), where α and β are any real numbers.

Key concepts

Linear Algebra

Linear algebra is a fundamental area of mathematics that focuses on the study of vectors, vector spaces, and linear transformations. It plays a key role in understanding eigensystems, which consist of eigenvalues and eigenvectors. In simple terms, linear algebra allows us to analyze and decompose matrices, helping us understand how transformations operate on vectors within a vector space. It's essential in various applications including engineering, physics, and data science. One core application of linear algebra is finding and interpreting eigenvalues and eigenvectors of a matrix. When a linear transformation is applied to a vector represented by a matrix, if the transformation only scales the vector without changing its direction, then such a vector is called an eigenvector, and the scale factor is known as the eigenvalue. Understanding these concepts builds a foundation for more complex topics such as machine learning, optimization, and dynamical systems. The ability to find eigenvalues and eigenvectors is crucial for simplifying matrix powers and solving differential equations.

Matrix Theory

Matrix theory extends the study of linear algebra into more complex realms by focusing on matrix operations, properties, and applications. A matrix is essentially a rectangular array of numbers or functions arranged in rows and columns, and they are used to represent linear transformations. Matrices are powerful tools used for solving systems of linear equations, as well as for performing operations like addition, subtraction, and multiplication. In the context of eigenvalues and eigenvectors, matrices are used to represent the linear transformation we are investigating. Matrix theory helps us understand essential concepts like: - **Diagonalization**: Transforming a matrix into a diagonal form, making it simpler to work with. - **Inverses**: Finding a matrix that reverses another matrix's transformation effects. - **Transpose**: Flipping a matrix over its diagonal. A key task in matrix theory is finding the determinant of a matrix, particularly when studying eigenvalues and eigenvectors. It helps us determine the invertibility of a matrix and other properties.

Determinant Calculation

Determinant calculation is a fundamental process in linear algebra and matrix theory. It involves obtaining a scalar value from a square matrix, which provides crucial information about the matrix.The determinant of a matrix is used to determine:- Whether a matrix is invertible, or non-singular (a non-zero determinant means it is invertible).- The volume distortion factor when the matrix represents a linear transformation.In the context of finding eigenvalues, determinant calculation is vital. To find eigenvalues, you compute the determinant of the matrix subtracted by \( \lambda\) times the identity matrix (\( A - \lambda I \)). Setting this determinant to zero, \( |A - \lambda I| = 0 \), allows us to solve for the eigenvalues \( \lambda \). This equation, known as the characteristic equation, is fundamental for solving for eigenvalues, which tells us how a matrix scales vectors. Calculating determinants accurately is essential; it confirms the existence and nature of eigenvalues and aids in various engineering, physics, and economics applications.