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Consider the matrix V E R"Xn where V has the structure V = I – 2vvT, the identity matrix I E R"X", and the vector v E R". The matrix V depends only on the identity matrix and vector v # 0, and occurs frequently in scientific computing and optimization applications. Task: Can you design an orthogonal matrix V? That is, if possible, pose a condition on the vector v / what property does v need to have, such that V = I – 2vv" is an orthogonal matrix? (v cannot be the zero vector, v70)

Consider the matrix V E R"Xn where V has the structure V = I – 2vvT, the identity matrix I E R"X", and the vector v E R". The matrix V depends only on the identity matrix and vector v # 0, and occurs frequently in scientific computing and optimization applications. Task: Can you design an orthogonal matrix V? That is, if possible, pose a condition on the vector v / what property does v need to have, such that V = I – 2vv" is an orthogonal matrix? (v cannot be the zero vector, v70)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.6: Matrices
Problem 17E
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Consider the matrix V E R"xn where V has the structure V = I – 2vvT, the identity matrix I E R"Xn, and the vector v E R". The matrix V depends only on the identity matrix and vector v # 0, and occurs frequently in scientific computing and optimization applications. Task: Can you design an orthogonal matrix V? That is, if possible, pose a condition on the vector v / what property does v need to have, such that V = I – 2vvT is an orthogonal matrix? (v cannot be the zero vector, v # 0)
Transcribed Image Text:Consider the matrix V E R"xn where V has the structure V = I – 2vvT, the identity matrix I E R"Xn, and the vector v E R". The matrix V depends only on the identity matrix and vector v # 0, and occurs frequently in scientific computing and optimization applications. Task: Can you design an orthogonal matrix V? That is, if possible, pose a condition on the vector v / what property does v need to have, such that V = I – 2vvT is an orthogonal matrix? (v cannot be the zero vector, v # 0)
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