Maximum Singular Value of $textbf{A} -textbf{B}$ for a Certain $textbf{B}$
up vote
1
down vote
favorite
Let $textbf{A} in mathbb{C}^{n times n} $, such that $rank(textbf{A}) = r$ and the singular values of $textbf{A}$ be $sigma_{1} geq dots geq sigma_{r} > 0$.
Let $textbf{u}_j$ and $textbf{v}_j$ for $1 geq j geq r$ denote the left and right singular vectors of $textbf{A}$ respectively.
Now, suppose we have a matrix
$$textbf{B} = alphasum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Where $alpha in mathbb{C}$ and $k < r$.
I want to find $|textbf{A} - textbf{B} |_{2}$, which is equivalent to the maximum singular value of $textbf{A} - textbf{B}$.
$textbf{Proof Attempt}$
The obvious move to me would be writing $textbf{A}$ as a sum of outer products.
$$textbf{A} = sum_{j=1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Rewriting $textbf{A} - textbf{B}$ gives
$$textbf{A} - textbf{B} = (1 - alpha)sum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*} + sum_{j=k+1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
This is where I get stuck. Because $alpha$ is an arbitrary complex scalar, I don't know what can be said about how it affects the singular values in the first sum, since they need to be both real and non-negative.
linear-algebra matrices complex-numbers singularvalues
add a comment |
up vote
1
down vote
favorite
Let $textbf{A} in mathbb{C}^{n times n} $, such that $rank(textbf{A}) = r$ and the singular values of $textbf{A}$ be $sigma_{1} geq dots geq sigma_{r} > 0$.
Let $textbf{u}_j$ and $textbf{v}_j$ for $1 geq j geq r$ denote the left and right singular vectors of $textbf{A}$ respectively.
Now, suppose we have a matrix
$$textbf{B} = alphasum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Where $alpha in mathbb{C}$ and $k < r$.
I want to find $|textbf{A} - textbf{B} |_{2}$, which is equivalent to the maximum singular value of $textbf{A} - textbf{B}$.
$textbf{Proof Attempt}$
The obvious move to me would be writing $textbf{A}$ as a sum of outer products.
$$textbf{A} = sum_{j=1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Rewriting $textbf{A} - textbf{B}$ gives
$$textbf{A} - textbf{B} = (1 - alpha)sum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*} + sum_{j=k+1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
This is where I get stuck. Because $alpha$ is an arbitrary complex scalar, I don't know what can be said about how it affects the singular values in the first sum, since they need to be both real and non-negative.
linear-algebra matrices complex-numbers singularvalues
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $textbf{A} in mathbb{C}^{n times n} $, such that $rank(textbf{A}) = r$ and the singular values of $textbf{A}$ be $sigma_{1} geq dots geq sigma_{r} > 0$.
Let $textbf{u}_j$ and $textbf{v}_j$ for $1 geq j geq r$ denote the left and right singular vectors of $textbf{A}$ respectively.
Now, suppose we have a matrix
$$textbf{B} = alphasum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Where $alpha in mathbb{C}$ and $k < r$.
I want to find $|textbf{A} - textbf{B} |_{2}$, which is equivalent to the maximum singular value of $textbf{A} - textbf{B}$.
$textbf{Proof Attempt}$
The obvious move to me would be writing $textbf{A}$ as a sum of outer products.
$$textbf{A} = sum_{j=1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Rewriting $textbf{A} - textbf{B}$ gives
$$textbf{A} - textbf{B} = (1 - alpha)sum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*} + sum_{j=k+1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
This is where I get stuck. Because $alpha$ is an arbitrary complex scalar, I don't know what can be said about how it affects the singular values in the first sum, since they need to be both real and non-negative.
linear-algebra matrices complex-numbers singularvalues
Let $textbf{A} in mathbb{C}^{n times n} $, such that $rank(textbf{A}) = r$ and the singular values of $textbf{A}$ be $sigma_{1} geq dots geq sigma_{r} > 0$.
Let $textbf{u}_j$ and $textbf{v}_j$ for $1 geq j geq r$ denote the left and right singular vectors of $textbf{A}$ respectively.
Now, suppose we have a matrix
$$textbf{B} = alphasum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Where $alpha in mathbb{C}$ and $k < r$.
I want to find $|textbf{A} - textbf{B} |_{2}$, which is equivalent to the maximum singular value of $textbf{A} - textbf{B}$.
$textbf{Proof Attempt}$
The obvious move to me would be writing $textbf{A}$ as a sum of outer products.
$$textbf{A} = sum_{j=1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
Rewriting $textbf{A} - textbf{B}$ gives
$$textbf{A} - textbf{B} = (1 - alpha)sum_{j=1}^{k}textbf{u}_j sigma_{j} textbf{v}_j^{*} + sum_{j=k+1}^{r}textbf{u}_j sigma_{j} textbf{v}_j^{*}$$
This is where I get stuck. Because $alpha$ is an arbitrary complex scalar, I don't know what can be said about how it affects the singular values in the first sum, since they need to be both real and non-negative.
linear-algebra matrices complex-numbers singularvalues
linear-algebra matrices complex-numbers singularvalues
asked Nov 17 at 16:07
Andreu Payne
304
304
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002513%2fmaximum-singular-value-of-textbfa-textbfb-for-a-certain-textbfb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown