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Math Help - Range of a Matrix

  1. #1
    uli
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    Range of a Matrix

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

    My Question is, how to obtain E ??

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

    And E is then the column space of C ??

    And when I want to find a matrix which is orthogonal to E, is it then a matrix which is orthogonal to 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) A(x) > 0 where A(x) = \sum_i x_i A_i + B and A_i = A_i^T and A_i , B is given. x is a vector in R^n.


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

    Tonio


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

    My Question is, how to obtain E ??

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

    And E is then the column space of C ??

    And when I want to find a matrix which is orthogonal to E, is it then a matrix which is orthogonal to 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 x = [x_1 x_2 ...]^T contains the decision variables x_i, which are scalar, so that the matrix 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 A_i and B are symmetric matrices with real elements.

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

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

    and then, how to get E ??
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