Question

Berechnen Sie die Eigenwerte, Eigenvektoren und gegebenfalls Hauptvektoren der Matrizen $$ A=\left(\begin{array}{llr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 3 & -3 \end{array}\right), B=\left(\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right), C=\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$

Short answer

Eigenvalues of A are 1, -1, -1. Eigenvectors are units of their free variables. B has eigenvalues 3, 0, 0 and eigenvectors [1, 1, 1], [-1, 1, 0], [-1, 0, 1]. C has eigenvalue 1 with the standard basis vectors as eigenvectors.

Step-by-step solution

Calculate Eigenvalues for Matrix A

To find the eigenvalues of matrix \( A \), we solve the characteristic equation \( \det(A - \lambda I) = 0 \). For \( \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 3 & -3 \end{bmatrix} \), its characteristic equation is:\[\det \begin{bmatrix} -\lambda & 1 & 0 \ 0 & -\lambda & 1 \ 1 & 3 & -3 - \lambda \end{bmatrix} = 0.\]Expand the determinant to obtain: \[-\lambda(-\lambda(-3-\lambda) - 1 \times 3) = 0.\]Solving \(-\lambda^3 + 3\lambda^2 - 3 = 0\) gives the eigenvalues: \( \lambda_1 = 1 \), \( \lambda_2 = \lambda_3 = -1 \).

Calculate Eigenvectors for Matrix A

For each eigenvalue \( \lambda \), solve \( (A - \lambda I) \mathbf{v} = 0 \) where \( \mathbf{v} \) is the eigenvector.**Eigenvalue \( \lambda_1 = 1 \):** \( (A - I) = \begin{bmatrix} -1 & 1 & 0 \ 0 & -1 & 1 \ 1 & 3 & -4 \end{bmatrix} \).Row reduce to find \( \mathbf{v}_1 \), e.g., \( \mathbf{v}_1 = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} \).**Eigenvalue \( \lambda_2 = \lambda_3 = -1 \):**The matrix \( (A + I) \) is \( \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \ 1 & 3 & -2 \end{bmatrix} \).Row reduce for solutions where \( \mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix} \) and a generalized eigenvector \( \mathbf{v}_3 = \begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix} \).

Calculate Eigenvalues for Matrix B

For \( \mathbf{B} = \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{bmatrix} \), the characteristic equation is formed as \( \det(B - \lambda I) = 0 \):\[\det \begin{bmatrix} 1 - \lambda & 1 & 1 \ 1 & 1 - \lambda & 1 \ 1 & 1 & 1 - \lambda \end{bmatrix} = 0.\]This reduces to: \( (1-\lambda) \left((1-\lambda)^2 - 0\right) = - \lambda^3 + 3\lambda^2 - 3\lambda + 1 \).Solving this, we get \( \lambda_1 = 3 \), \( \lambda_2 = \lambda_3 = 0 \).

Calculate Eigenvectors for Matrix B

For \( \lambda_1 = 3 \) and \( (B - 3I)\mathbf{v} = 0 \), matrix becomes \( \begin{bmatrix} -2 & 1 & 1 \ 1 & -2 & 1 \ 1 & 1 & -2 \end{bmatrix} \).Solving, the eigenvector \( \mathbf{u} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} \).For \( \lambda_2 = 0 \), solving \( B\mathbf{v} = 0 \), choose independent vectors:\( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) such as \( \begin{bmatrix} -1 \ 1 \ 0 \end{bmatrix} \) and \( \begin{bmatrix} -1 \ 0 \ 1 \end{bmatrix} \).

Calculate Eigenvalues for Matrix C

Matrix \( \mathbf{C} = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \) is the identity, so all eigenvalues are \( \lambda = 1 \).

Calculate Eigenvectors for Matrix C

For the identity matrix, any unit vector can be an eigenvector for \( \lambda = 1 \). Typical choices can be:\( \mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \ 0 \ 0 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \ 0 \ 0 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} 0 \ 0 \ 1 \ 0 \end{bmatrix}, \mathbf{v}_4 = \begin{bmatrix} 0 \ 0 \ 0 \ 1 \end{bmatrix}. \)

Key concepts

Eigenvalues

Eigenvalues are crucial in linear algebra as they represent the special numbers associated with a square matrix. When dealing with matrices, these numbers reveal much about the matrix's properties.
To find eigenvalues, you solve the characteristic equation given by \( \det(A - \lambda I) = 0 \), where \( A \) is your matrix and \( I \) is the identity matrix of the same size. Eigenvalues tell you about the matrix's scaling factors along certain directions.
This concept becomes particularly useful in solving systems of linear equations and understanding transformations.

• Provide insights into the dimensions transformed or scaled by the matrix.
• Aid in simplifying matrices for easier computation and analysis.
• Help in calculating the determinant and trace of matrices, leading to vital applications in physics and computer science.

Eigenvectors

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation (represented by a matrix) is applied.
In practical terms, if you apply matrix \( A \) to eigenvector \( \mathbf{v} \), \( A\mathbf{v} \), the result is a multiple of \( \mathbf{v} \).

Finding eigenvectors involves solving the equation \( (A - \lambda I)\mathbf{v} = 0 \) for each eigenvalue \( \lambda \). Only non-zero solutions are considered as valid eigenvectors. They help identify directions in which the transformation remains consistent.

• Indicate invariant directions in the transformation represented by the matrix.
• Useful in diagonalizing matrices which simplifies matrix operations.
• ARE essential in quantum mechanics, vibrations analysis, and principal component analysis in statistics.

Characteristic Equation

The characteristic equation is a polynomial equation derived from a square matrix, crucial for finding a matrix's eigenvalues.
To form it, substitute the eigenvalues \( \lambda \) into the equation \( \det(A - \lambda I) = 0 \). This involves calculating the determinant of the matrix \( A - \lambda I \), which results in a polynomial in terms of \( \lambda \).
The roots of this polynomial reflect the eigenvalues of the matrix. Understanding the characteristic equation is fundamental to solving matrix-related problems in linear algebra.

• Provides the polynomial whose roots are the eigenvalues.
• Allows determination of the nature and stability of systems in control theory.
• Crucial in various applications, like analyzing stability in differential equations.

Matrix

A matrix is a rectangular array of numbers or expressions arranged in rows and columns that is fundamental to linear algebra.
Matrices are used to represent and solve systems of linear equations, and they are foundational in transformation operations in vector spaces.
Understanding matrix operations, like addition, multiplication, and finding the determinant, are essential to manipulating these arrays in applications.

• Represent linear transformations and systems of equations efficiently.
• Can be used to perform complex calculations and transformations in graphics.
• Are the backbone of algorithms in machine learning and computer graphics.

Identity Matrix

The identity matrix, denoted as \( I \), is a special type of matrix where all the diagonal elements are 1, and all other elements are 0.
When we multiply any matrix by the identity matrix, the original matrix is unchanged. It's equivalent to multiplying a number by 1 in arithmetic.
This concept is vital in linear algebra for defining the characteristic equation and serves as the neutral element in matrix algebra.

• Acts like the number 1 in matrix multiplication (preserves original matrix).
• Used in finding inverses of matrices, which is crucial for solving systems of equations.
• Essential for defining eigenvalues and eigenvectors as part of the characteristic equation.