Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n x n matrix A is grumpy if it has n distinct negative" eigenvalues. "ie. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: -1 -2 is grumpy, as it has eigenvalues –1 and -3. -3 The matrix 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 Def
Read the following definition and example carefully, and then answer the question below. We say (for n > 2) that an n x n matrix A is grumpy if it has n distinct negative" eigenvalues. "ie. if A is an eigenvalue of A, then A < 0. Here is an example and a couple quick consequences of the definition: -1 -2 is grumpy, as it has eigenvalues –1 and -3. -3 The matrix 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 Def
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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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