Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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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.
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