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1. Consider the matrices A₁ 4 = ( 1 ) - 1 3 0 A₂ 42 = (3-2) ₁ 4₁ = ( 2² -2²), , A3 0 -2 These matrices can be used to build functions x¹A₁x, x¹ A₂x, x¹ A3x, and x¹ A4r. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. Explain briefly. 2. Consider matrices of the form (a) Ma, ≥ 0. (b) Ma,B 0. 3. Is the matrix (b) Figure 1: Graphical representation of functions Ma,B := a 1 1 = a (-2² -²2 ) + 8 (-1 -2 ) where a and 3 are scalars. Depending on the values of a and 3, these matrices might be psd, nsd, or indefinite. In a graph with axes a and 3, represent the set of all (a, 3) values for which the matrix 1 5 -4 1 psd, pd, nsd, nd, or indefinite? Justify your answer. 4. Consider the matrix -a 0 0 and A4 = ( 1 5 -1 4 -(-6). -4 -a (a² + a) 0 0 0 0 -2a (2-a²) (2-a²) -2a (2-a²) 2+4aa² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd? €

1. Consider the matrices A₁ 4 = ( 1 ) - 1 3 0 A₂ 42 = (3-2) ₁ 4₁ = ( 2² -2²), , A3 0 -2 These matrices can be used to build functions x¹A₁x, x¹ A₂x, x¹ A3x, and x¹ A4r. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. Explain briefly. 2. Consider matrices of the form (a) Ma, ≥ 0. (b) Ma,B 0. 3. Is the matrix (b) Figure 1: Graphical representation of functions Ma,B := a 1 1 = a (-2² -²2 ) + 8 (-1 -2 ) where a and 3 are scalars. Depending on the values of a and 3, these matrices might be psd, nsd, or indefinite. In a graph with axes a and 3, represent the set of all (a, 3) values for which the matrix 1 5 -4 1 psd, pd, nsd, nd, or indefinite? Justify your answer. 4. Consider the matrix -a 0 0 and A4 = ( 1 5 -1 4 -(-6). -4 -a (a² + a) 0 0 0 0 -2a (2-a²) (2-a²) -2a (2-a²) 2+4aa² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd? €

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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1. Consider the matrices 4 A₁ = - (13) 42-(3-2) 0 23-2). These matrices can be used to build functions x¹A₁x, x¹ A₂x, x¹ A3x, and x¹ A4x. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. = Explain briefly. 2. Consider matrices of the form (a) Ma, ≥ 0. (b) Ma,ß ≤ 0. 3. Is the matrix Ma,8:= a 0 (b) (c) Figure 1: Graphical representation of functions , A3 = 1 -a 0 0 1 -2 α ( -2 -² ) + ³ ( -1 -₂ ) 2 -2 where a and 3 are scalars. Depending on the values of a and 3, these matrices might be psd, nsd, or indefinite. In a graph with axes a and 3, represent the set of all (a, 3) values for which the matrix 5 -a (a² + a) 0 -4 psd, pd, nsd, nd, or indefinite? Justify your answer. 4. Consider the matrix 1 " 1 5 -1 4 and A4 = -5 -4 0 0 0 -2a (2-a²) -2a (2 a²) - (2-a²) 2+ 4a - a² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd? ə
Transcribed Image Text:1. Consider the matrices 4 A₁ = - (13) 42-(3-2) 0 23-2). These matrices can be used to build functions x¹A₁x, x¹ A₂x, x¹ A3x, and x¹ A4x. These functions are represented (in an arbitrary order) in Figure 1. Match each matrix with its corresponding picture. = Explain briefly. 2. Consider matrices of the form (a) Ma, ≥ 0. (b) Ma,ß ≤ 0. 3. Is the matrix Ma,8:= a 0 (b) (c) Figure 1: Graphical representation of functions , A3 = 1 -a 0 0 1 -2 α ( -2 -² ) + ³ ( -1 -₂ ) 2 -2 where a and 3 are scalars. Depending on the values of a and 3, these matrices might be psd, nsd, or indefinite. In a graph with axes a and 3, represent the set of all (a, 3) values for which the matrix 5 -a (a² + a) 0 -4 psd, pd, nsd, nd, or indefinite? Justify your answer. 4. Consider the matrix 1 " 1 5 -1 4 and A4 = -5 -4 0 0 0 -2a (2-a²) -2a (2 a²) - (2-a²) 2+ 4a - a² Find all values of a for which this matrix is psd. Are there values of a for which the matrix is pd? ə
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