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Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n xn matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A <0. Here is an example and a couple quick consequences of the definition: -1 The matrix is grumpy, as it has eigenvalues -1 and -3. All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Def

Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n xn matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A <0. Here is an example and a couple quick consequences of the definition: -1 The matrix is grumpy, as it has eigenvalues -1 and -3. All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Def

Advanced Engineering Mathematics
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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Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n xn matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: The matrix is grumpy, as it has eigenvalues -1 and -3. All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Thm
Transcribed Image Text:Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n xn matrix A is grumpy if it has n distinct negative" eigenvalues. "i.e. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: The matrix is grumpy, as it has eigenvalues -1 and -3. All grumpy matrices are diagonalizable (since they have n distinct eigenvalues.) Similarly, all grumpy matrices are also invertible (since they cannot have 0 as an eigenvalue.) Prove that if A is a 2 x 2 grumpy matrix,' then det(A) is equal to the area of the fundamental parallelogram of TA (i.e. the image of the unit square under the linear transformation TA). Thm
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