Let P1, P2 E R"X" be two projection matrices (so that Pi = P; , P{ = P¡) with P1P2 = 0. Let the rank of P; be r¡ (so that we must have r1 + r2
Let P1, P2 E R"X" be two projection matrices (so that Pi = P; , P{ = P¡) with P1P2 = 0. Let the rank of P; be r¡ (so that we must have r1 + r2
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 34E
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![Let P1, P2 E R"*" be two projection matrices (so that Pi = P; , P; = P;) with P1P2 = 0. Let the rank
of P; be r; (so that we must have ri + r2 < n). Let D1 = diag(1,, ,0-r, ), D2 = diag(0,", , 1,,,0,-r,-r2) €
R"×n be two diagonal matrices with diagonal elements in {0,1}. We would like to show that, there exists
an orthogonal matrix U € R"xn, such that Pı = UD¡U™ and P2 = UD2U' (i.e., P1 and P2 are
simultaneously diagonalizable). To show this, one can proceed in the following way.
||
1. Show that there exists orthogonal matrices V1, V2 € R"×n, such that P1 = V1D¡V} and P2 =
V½D½V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1).
2. Let U1 E R"×"1 be a submatrix of V1 E R"Xn by selecting the first rị columns of V1. Let U2 E R"×r2
be a submatrix of V2 E R"×n by selecting the ri+1 to r1+r2 columns of V 2. Show that P1 =U‚U
and P2 = U,Ū,.
3. Show that UU2 = 0,1xr2 (use the properties that P¡P2 = 0, U; U; = I,,).
.
4. Show that there exists U3 € R"×(n-ri-r2), such that if we define U
is an orthogonal matrix.
[U1,U2,U3] E R"×n, then U
5. Show that Pı = UD¸U' and P2 = UD¡U'. Then this U matrix is the U matrix we would like
find.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdca5b43e-59da-44fb-bdbe-c77e0d2759bd%2F73ab3363-9ab4-433c-a62d-afc1fecd6354%2F70gzkfa_processed.png&w=3840&q=75)
Transcribed Image Text:Let P1, P2 E R"*" be two projection matrices (so that Pi = P; , P; = P;) with P1P2 = 0. Let the rank
of P; be r; (so that we must have ri + r2 < n). Let D1 = diag(1,, ,0-r, ), D2 = diag(0,", , 1,,,0,-r,-r2) €
R"×n be two diagonal matrices with diagonal elements in {0,1}. We would like to show that, there exists
an orthogonal matrix U € R"xn, such that Pı = UD¡U™ and P2 = UD2U' (i.e., P1 and P2 are
simultaneously diagonalizable). To show this, one can proceed in the following way.
||
1. Show that there exists orthogonal matrices V1, V2 € R"×n, such that P1 = V1D¡V} and P2 =
V½D½V, (hint: use the properties that the eigenvalues of projection matrices are either 0 or 1).
2. Let U1 E R"×"1 be a submatrix of V1 E R"Xn by selecting the first rị columns of V1. Let U2 E R"×r2
be a submatrix of V2 E R"×n by selecting the ri+1 to r1+r2 columns of V 2. Show that P1 =U‚U
and P2 = U,Ū,.
3. Show that UU2 = 0,1xr2 (use the properties that P¡P2 = 0, U; U; = I,,).
.
4. Show that there exists U3 € R"×(n-ri-r2), such that if we define U
is an orthogonal matrix.
[U1,U2,U3] E R"×n, then U
5. Show that Pı = UD¸U' and P2 = UD¡U'. Then this U matrix is the U matrix we would like
find.
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