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Thread: Range of a Matrix

  1. #1
    uli
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    Range of a Matrix

    A Linear Matrix Inequality (LMI) $\displaystyle A(x) > 0$ where $\displaystyle A(x) = \sum_i x_i A_i + B$ and $\displaystyle A_i = A_i^T$ and $\displaystyle A_i , B$ is given. $\displaystyle x$ is a vector in $\displaystyle R^n$.
    Then, the linear map $\displaystyle x \rightarrow Cx$ from $\displaystyle R^n$ to $\displaystyle S$, where $\displaystyle S$ is a space of symmetric matrices with prescribed block diagonal structure, is associated such that $\displaystyle A(x) = Cx+B$. The range of this map is $\displaystyle E = range(C)$.

    My Question is, how to obtain $\displaystyle E$ ??

    I guess, that $\displaystyle C = [A_1,A_2,...]$ is a matrix which columns are the matrices $\displaystyle A_i$.

    And $\displaystyle E$ is then the column space of $\displaystyle C$ ??

    And when I want to find a matrix which is orthogonal to $\displaystyle E$, is it then a matrix which is orthogonal to $\displaystyle C$ ??

    As you might have noticed, I've no idea what to do.
    So, please help me.

    thanks
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  2. #2
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    Quote Originally Posted by uli View Post
    A Linear Matrix Inequality (LMI) $\displaystyle A(x) > 0$ where $\displaystyle A(x) = \sum_i x_i A_i + B$ and $\displaystyle A_i = A_i^T$ and $\displaystyle A_i , B$ is given. $\displaystyle x$ is a vector in $\displaystyle R^n$.


    ...and thus $\displaystyle Ax$ is a vector in $\displaystyle \mathbb{R}^n$. What does it mean that a vector is greater/less than zero?

    Tonio


    Then, the linear map $\displaystyle x \rightarrow Cx$ from $\displaystyle R^n$ to $\displaystyle S$, where $\displaystyle S$ is a space of symmetric matrices with prescribed block diagonal structure, is associated such that $\displaystyle A(x) = Cx+B$. The range of this map is $\displaystyle E = range(C)$.

    My Question is, how to obtain $\displaystyle E$ ??

    I guess, that $\displaystyle C = [A_1,A_2,...]$ is a matrix which columns are the matrices $\displaystyle A_i$.

    And $\displaystyle E$ is then the column space of $\displaystyle C$ ??

    And when I want to find a matrix which is orthogonal to $\displaystyle E$, is it then a matrix which is orthogonal to $\displaystyle C$ ??

    As you might have noticed, I've no idea what to do.
    So, please help me.

    thanks
    .
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  3. #3
    uli
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    Quote Originally Posted by tonio View Post
    ...and thus is a vector in . What does it mean that a vector is greater/less than zero?

    Tonio
    the vector $\displaystyle x = [x_1 x_2 ...]^T$ contains the decision variables $\displaystyle x_i$, which are scalar, so that the matrix $\displaystyle A(x) = \sum_i x_i A_i + B$ is positive definite. So that it results in an optimization problem which can be solved by using i.e. Semidefinite programming, what I am trying to do.
    The matrices $\displaystyle A_i$ and $\displaystyle B$ are symmetric matrices with real elements.

    At the time, I stuck with finding $\displaystyle E = range(C)$ and $\displaystyle x \mapsto Cx$ such that $\displaystyle A(x) = Cx + B$.

    How to obtain $\displaystyle C$ and remains the vector $\displaystyle x$ a vector or becomes $\displaystyle x$ a matrix after mapping since $\displaystyle Cx$ must be of same dimension as $\displaystyle B$ because of adding this two matrices ??

    and then, how to get $\displaystyle E$ ??
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