Mixing
Frobenius norm equality related to orthogonal projections
0
$begingroup$
Let $k > 1$. For which $v_1, v_2, dots, v_k in mathbb{R}^{n times 1}$ with $Vert v_i Vert = 1$ holds the following equation
$$
leftVert prod_{i=1}^k v_i v_i^T rightVert_F = leftVert prod_{i=1}^k (I - v_i v_i^T) rightVert_F,
$$
where $Vert Vert_F$ indicates the Frobenius norm and $I$ is the identity matrix?
linear-algebra
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
$endgroup$
|
show 2 more comments
0
$begingroup$
Let $k > 1$. For which $v_1, v_2, dots, v_k in mathbb{R}^{n times 1}$ with $Vert v_i Vert = 1$ holds the following equation
$$
leftVert prod_{i=1}^k v_i v_i^T rightVert_F = leftVert prod_{i=1}^k (I - v_i v_i^T) rightVert_F,
$$
where $Vert Vert_F$ indicates the Frobenius norm and $I$ is the identity matrix?
linear-algebra
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
$endgroup$
$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
1
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
1
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55
|
show 2 more comments
0
0
0
$begingroup$
Let $k > 1$. For which $v_1, v_2, dots, v_k in mathbb{R}^{n times 1}$ with $Vert v_i Vert = 1$ holds the following equation
$$
leftVert prod_{i=1}^k v_i v_i^T rightVert_F = leftVert prod_{i=1}^k (I - v_i v_i^T) rightVert_F,
$$
where $Vert Vert_F$ indicates the Frobenius norm and $I$ is the identity matrix?
linear-algebra
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
$endgroup$
Let $k > 1$. For which $v_1, v_2, dots, v_k in mathbb{R}^{n times 1}$ with $Vert v_i Vert = 1$ holds the following equation
$$
leftVert prod_{i=1}^k v_i v_i^T rightVert_F = leftVert prod_{i=1}^k (I - v_i v_i^T) rightVert_F,
$$
where $Vert Vert_F$ indicates the Frobenius norm and $I$ is the identity matrix?
linear-algebra
linear-algebra
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
edited Jan 15 at 19:42
Sebastian Schlecht
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
asked Jan 15 at 19:27
Sebastian SchlechtSebastian Schlecht
23018
23018
$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
1
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
1
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55
|
show 2 more comments
$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
1
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
1
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55
$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
1
1
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
1
1
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55
|
show 2 more comments
0
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$begingroup$
A Householder reflection has the form $I-color{red}2 VV^T$
$endgroup$
– Jean Marie
Jan 15 at 19:36
$begingroup$
@JeanMarie I am aware of this. However, this is no typo. The original problem is derived from a Householder matrix. Maybe at this point, its not a good name anymore, I agree.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:37
$begingroup$
$I-VV^T$ is the matrix of the orthogonal projection onto the hyperplane with normal vector $V$, whereas $VV^T$ (rank one matrix) is the matrix of the orthogonal projection onto vector $V$.
$endgroup$
– Jean Marie
Jan 15 at 19:39
1
$begingroup$
@JeanMarie thanks for the explanation. I changed the title accordingly.
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:42
1
$begingroup$
@JeanMarie right, and $k leq 1$
$endgroup$
– Sebastian Schlecht
Jan 15 at 19:55