Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 36
a. Prove that the eigenvalues of an \(n \times n\) real matrix \(A\) are the same as the eigenvalues of \(A^{T}\). b. With reference to part (a), show by a counterexample that an eigenvector of \(A\) need not be an eigenvector of \(A^{T}\).
Short Answer
Expert verified
Eigenvalues of \(A\) and \(A^T\) are the same, but eigenvectors can differ; \(A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) is a counterexample.
Step by step solution
01
Define Eigenvalues
An eigenvalue of a matrix \(A\) is a scalar \(\lambda\) such that there exists a non-zero vector \(\mathbf{x}\) satisfying the equation \(A\mathbf{x} = \lambda \mathbf{x}\).
02
Define Characteristic Polynomial
The eigenvalues of a matrix \(A\) are the roots of its characteristic polynomial, which is given by \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix.
03
Relation with Transpose
For the purposes of comparing eigenvalues, we note that \(A\) and its transpose \(A^T\) have the same characteristic polynomial, because \(\det(A - \lambda I) = \det(A^T - \lambda I)\). This implies that \(A\) and \(A^T\) have the same eigenvalues.
04
Counterexample Setup
To demonstrate that an eigenvector of \(A\) is not necessarily an eigenvector of \(A^T\), consider a specific counterexample where this is the case, such as matrix \(A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\).
05
Eigenvector for A
For matrix \(A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}\), solving \(A\mathbf{x} = \lambda \mathbf{x}\) shows that \(\lambda = 0\) and a corresponding eigenvector is \(\mathbf{x} = \begin{bmatrix} 1 \ 0 \end{bmatrix}\).
06
Eigenvector for A^T
Now consider \(A^T = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}\). Solving \(A^T\mathbf{x} = \lambda \mathbf{x}\) shows that \(\lambda = 0\) and the corresponding eigenvector cannot be \(\begin{bmatrix} 1 \ 0 \end{bmatrix}\). Instead, it is \(\begin{bmatrix} 0 \ 1 \end{bmatrix}\).
07
Conclusion of Counterexample
This counterexample confirms that although \(A\) and \(A^T\) have the same eigenvalues, \(\begin{bmatrix} 1 \ 0 \end{bmatrix}\) is an eigenvector of \(A\) for \(\lambda=0\) but not of \(A^T\). Similarly, \(\begin{bmatrix} 0 \ 1 \end{bmatrix}\) is an eigenvector of \(A^T\) for \(\lambda=0\) but not of \(A\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eigenvectors
An eigenvector is a special vector associated with a matrix. It is non-zero and when the matrix acts on it, the vector is simply scaled by a factor called the eigenvalue. This is mathematically represented as:
- For a matrix \( A \) and scalar \( \lambda \), a vector \( \mathbf{x} \) is an eigenvector if \( A\mathbf{x} = \lambda \mathbf{x} \).
- The eigenvector does not change direction under the transformation described by the matrix.
- Eigenvectors are only defined up to a scalar multiple; multiplying an eigenvector by any non-zero scalar yields another eigenvector.
Real Matrix
A real matrix is a matrix whose entries are all real numbers. It plays an important role in various fields such as physics, engineering, and statistics.
- A matrix is defined by its dimensions, indicated as \( m \times n \), where \( m \) and \( n \) are the number of rows and columns, respectively.
- In the context of eigenvalues and eigenvectors, real matrices are particularly interesting because their properties allow for real eigenvalues and corresponding eigenvectors, although complex pairs may also occur.
- The behavior of real matrices helps predict dynamic systems, solve differential equations, and transform geometric spaces, making them fundamental in both theoretical and applied mathematics.
Transpose of a Matrix
The transpose of a matrix \( A \), denoted as \( A^T \), is formed by flipping \( A \) over its diagonal. This means that the element at the \(i, j\)-th position in \( A \) becomes the element at the \(j, i\)-th position in \( A^T \).
- This operation is useful in various contexts, particularly in solving systems of linear equations, as it often helps in simplifying the equations involved.
- An important property of the transpose is that for any two matrices \( A \) and \( B \), the transpose of the product \( (AB)^T \) is equal to \( B^T A^T \).
- When considering eigenvalues, the transpose of a real square matrix has the same eigenvalues as the original matrix, due to having the same characteristic polynomial.
Characteristic Polynomial
The characteristic polynomial of a matrix \( A \) is an essential tool for finding its eigenvalues. It is a polynomial derived from the matrix by subtracting \( \lambda \) times the identity matrix from \( A \), and then taking the determinant.
- Expressed as \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix, this polynomial's roots are the eigenvalues of the matrix.
- For a \( n \times n \) matrix, the characteristic polynomial is of degree \( n \), thus there can be up to \( n \) eigenvalues, considering complex numbers and their multiplicities.
- The characteristic polynomial is pivotal in proving the relations between eigenvalues of a matrix and its transpose.
