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. ### 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.

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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

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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.

### 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.

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