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Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in %3D R", let ŷ = By and z = y – ŷ. a. Show that z is orthogonal to ŷ. b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in w1. Why does this prove that By is the orthogonal projection of y onto the column space of B?

Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in %3D R", let ŷ = By and z = y – ŷ. a. Show that z is orthogonal to ŷ. b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in w1. Why does this prove that By is the orthogonal projection of y onto the column space of B?

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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Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in %3D R", let ŷ = By and z = y – ŷ. a. Show that z is orthogonal to ŷ. b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in w1. Why does this prove that By is the orthogonal projection of y onto the column space of B?
Transcribed Image Text:Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any y in %3D R", let ŷ = By and z = y – ŷ. a. Show that z is orthogonal to ŷ. b. Let W be the column space of B. Show that y is the sum of a vector in W and a vector in w1. Why does this prove that By is the orthogonal projection of y onto the column space of B?
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