The eigenvalues of a triangular matrix are on the diagonal. A diagonal matrix is diagonalizable. Eigenvectors from distinct eigenvalues are linearly independent. There exists a matrix that is invertible but not diagonalizable.
The eigenvalues of a triangular matrix are on the diagonal. A diagonal matrix is diagonalizable. Eigenvectors from distinct eigenvalues are linearly independent. There exists a matrix that is invertible but not diagonalizable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Mark each statement T if the statement is always true or F if it’s ever false. Do not assume anything beyond what is explicitly stated.
Transcribed Image Text:### Linear Algebra Properties of Matrices
(e) **The eigenvalues of a triangular matrix are on the diagonal.**
- In a triangular matrix (whether upper or lower triangular), the eigenvalues are found directly on the diagonal elements. This property simplifies the calculation of eigenvalues in such matrices.
(f) **A diagonal matrix is diagonalizable.**
- A diagonal matrix is already in its simplest form, and thus it is inherently diagonalizable. A matrix is diagonalizable if it is similar to a diagonal matrix, meaning there exists a matrix \(P\) such that \(P^{-1}AP\) is diagonal.
(g) **Eigenvectors from distinct eigenvalues are linearly independent.**
- If a matrix has distinct eigenvalues, the eigenvectors corresponding to these distinct eigenvalues are linearly independent. This property is essential in understanding the dimensionality and the structure of eigenspaces.
(h) **There exists a matrix that is invertible but not diagonalizable.**
- It is possible for an invertible matrix to not be diagonalizable. This occurs, for example, when the matrix has defective eigenvalues (eigenvalues which do not have enough linearly independent eigenvectors to form a basis). One such example is a Jordan block matrix.
These properties are fundamental in the study and application of linear algebra, especially in understanding the behavior and transformations associated with matrices.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
