Show that there exists orthogonal matrices V1, V2 € R"×n, such that Pı = VịD¡V{ and P2 V½D¿V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1). %3D IDn xn hu gel Lot IT ID n xra
Show that there exists orthogonal matrices V1, V2 € R"×n, such that Pı = VịD¡V{ and P2 V½D¿V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1). %3D IDn xn hu gel Lot IT ID n xra
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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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Question
![1. Show that there exists orthogonal matrices V1, V2 € R"xn, such that P1 = VịD¡V; and P2 =
V½D½V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1).
2. Let Uj € R"×"1 be a submatrix of V1 E R"×n by selecting the first rị columns of V1. Let U2 E R"×r2
be a submatrix of V2 € R"×n by selecting the ri+1 to r1+r2 columns of V2. Show that Pı = U¡U|
and P2 = U2U¸.
-T
3. Show that U¡U2= 0,r1xr2 (use the properties that P1P2 = 0, U;U; =
= Ir,).
rixr2
= [U1,U2‚U3] € R"xn, then U
4. Show that there exists U3 € R"×(n-r1-r2), such that if we define U
is an orthogonal matrix.
5. Show that P1 = ŪD¸U' and P2 = UD¡U'. Then this U matrix is the U matrix we would like to
find.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdca5b43e-59da-44fb-bdbe-c77e0d2759bd%2F5abf24d6-2199-4424-b473-42cc9a2367af%2Fvak1kbl_processed.png&w=3840&q=75)
Transcribed Image Text:1. Show that there exists orthogonal matrices V1, V2 € R"xn, such that P1 = VịD¡V; and P2 =
V½D½V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1).
2. Let Uj € R"×"1 be a submatrix of V1 E R"×n by selecting the first rị columns of V1. Let U2 E R"×r2
be a submatrix of V2 € R"×n by selecting the ri+1 to r1+r2 columns of V2. Show that Pı = U¡U|
and P2 = U2U¸.
-T
3. Show that U¡U2= 0,r1xr2 (use the properties that P1P2 = 0, U;U; =
= Ir,).
rixr2
= [U1,U2‚U3] € R"xn, then U
4. Show that there exists U3 € R"×(n-r1-r2), such that if we define U
is an orthogonal matrix.
5. Show that P1 = ŪD¸U' and P2 = UD¡U'. Then this U matrix is the U matrix we would like to
find.
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