Prove statements with oblique projections
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I can not really understand how to prove the following statements in the task. I have found some information about oblique projections at the Wikipedia, but it is really complicated there and without any examples. Thanks for any hint in advance!
Task:
Assume that $mathbb{R}^n$ could be represented as the direct (not necessarily orthogonal) sum $M_1 dotplus M_2$ of two its subspaces $M_1$ and $M_2$. Particulary, every $x in mathbb{R}^n$ can be represented as $x = x_1 + x_2$ with some $x_j in M_j$, and the mapping $P_j: x mapsto x_j$ is called the (oblique) projector onto $M_j$ parallel to $M_{3-j}$. Then $P_j$ satisfies the following properties: $P_1 + P_2 = I_n$, $P_j^2 = P_j$ and $P_1P_2 = P_2 P_1 = 0$.
(a) Show that $P$ which satisfies $P^2 = P$ is a projector onto some subspace $L$ parallel to $M$. Identify these $L$ and $M$.
(b) Show that the projector $P$ is an orthogonal projector if and only if $P$ is symmetric.
(c) Assume that two transformations $P_1$ and $P_2$ of $mathbb{R}^n$ satisfy conditions: $P_1 + P_2 = I_n$ and $P_1P_2 = 0$. Prove that $P_1$ and $P_2$ are projectors and that $P_2P_1 = 0$.
projection
asked Dec 1 '18 at 15:37
MichaelMichael
1055
$endgroup$
add a comment |
$begingroup$
I can not really understand how to prove the following statements in the task. I have found some information about oblique projections at the Wikipedia, but it is really complicated there and without any examples. Thanks for any hint in advance!
Task:
Assume that $mathbb{R}^n$ could be represented as the direct (not necessarily orthogonal) sum $M_1 dotplus M_2$ of two its subspaces $M_1$ and $M_2$. Particulary, every $x in mathbb{R}^n$ can be represented as $x = x_1 + x_2$ with some $x_j in M_j$, and the mapping $P_j: x mapsto x_j$ is called the (oblique) projector onto $M_j$ parallel to $M_{3-j}$. Then $P_j$ satisfies the following properties: $P_1 + P_2 = I_n$, $P_j^2 = P_j$ and $P_1P_2 = P_2 P_1 = 0$.
(a) Show that $P$ which satisfies $P^2 = P$ is a projector onto some subspace $L$ parallel to $M$. Identify these $L$ and $M$.
(b) Show that the projector $P$ is an orthogonal projector if and only if $P$ is symmetric.
(c) Assume that two transformations $P_1$ and $P_2$ of $mathbb{R}^n$ satisfy conditions: $P_1 + P_2 = I_n$ and $P_1P_2 = 0$. Prove that $P_1$ and $P_2$ are projectors and that $P_2P_1 = 0$.
projection
asked Dec 1 '18 at 15:37
MichaelMichael
1055
$endgroup$
$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02
add a comment |
$begingroup$
I can not really understand how to prove the following statements in the task. I have found some information about oblique projections at the Wikipedia, but it is really complicated there and without any examples. Thanks for any hint in advance!
Task:
Assume that $mathbb{R}^n$ could be represented as the direct (not necessarily orthogonal) sum $M_1 dotplus M_2$ of two its subspaces $M_1$ and $M_2$. Particulary, every $x in mathbb{R}^n$ can be represented as $x = x_1 + x_2$ with some $x_j in M_j$, and the mapping $P_j: x mapsto x_j$ is called the (oblique) projector onto $M_j$ parallel to $M_{3-j}$. Then $P_j$ satisfies the following properties: $P_1 + P_2 = I_n$, $P_j^2 = P_j$ and $P_1P_2 = P_2 P_1 = 0$.
(a) Show that $P$ which satisfies $P^2 = P$ is a projector onto some subspace $L$ parallel to $M$. Identify these $L$ and $M$.
(b) Show that the projector $P$ is an orthogonal projector if and only if $P$ is symmetric.
(c) Assume that two transformations $P_1$ and $P_2$ of $mathbb{R}^n$ satisfy conditions: $P_1 + P_2 = I_n$ and $P_1P_2 = 0$. Prove that $P_1$ and $P_2$ are projectors and that $P_2P_1 = 0$.
projection
asked Dec 1 '18 at 15:37
MichaelMichael
1055
$endgroup$
I can not really understand how to prove the following statements in the task. I have found some information about oblique projections at the Wikipedia, but it is really complicated there and without any examples. Thanks for any hint in advance!
Task:
Assume that $mathbb{R}^n$ could be represented as the direct (not necessarily orthogonal) sum $M_1 dotplus M_2$ of two its subspaces $M_1$ and $M_2$. Particulary, every $x in mathbb{R}^n$ can be represented as $x = x_1 + x_2$ with some $x_j in M_j$, and the mapping $P_j: x mapsto x_j$ is called the (oblique) projector onto $M_j$ parallel to $M_{3-j}$. Then $P_j$ satisfies the following properties: $P_1 + P_2 = I_n$, $P_j^2 = P_j$ and $P_1P_2 = P_2 P_1 = 0$.
(a) Show that $P$ which satisfies $P^2 = P$ is a projector onto some subspace $L$ parallel to $M$. Identify these $L$ and $M$.
(b) Show that the projector $P$ is an orthogonal projector if and only if $P$ is symmetric.
(c) Assume that two transformations $P_1$ and $P_2$ of $mathbb{R}^n$ satisfy conditions: $P_1 + P_2 = I_n$ and $P_1P_2 = 0$. Prove that $P_1$ and $P_2$ are projectors and that $P_2P_1 = 0$.
projection
projection
asked Dec 1 '18 at 15:37
MichaelMichael
1055
asked Dec 1 '18 at 15:37
MichaelMichael
1055
asked Dec 1 '18 at 15:37
MichaelMichael
1055
asked Dec 1 '18 at 15:37
MichaelMichael
1055
asked Dec 1 '18 at 15:37
MichaelMichael
1055
1055
$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02
add a comment |
$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02
$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02
$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02
add a comment |
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$begingroup$
Have you looked at the effect of $P_j$ on $M_1$ and $M_2?$
$endgroup$
– saulspatz
Dec 1 '18 at 16:02