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Symmetric tridiagonal matrix

  1. Feb 8, 2009 #1
    The matrix A is symmetric and tridiagonal.
    If B is the matrix formed from A by deleting the first two rows and columns, show that [tex]\left|A\right|[/tex] = a[tex]_{}11[/tex][tex]\left|M_{}11\right|[/tex] - (a[tex]_{}1[/tex])[tex]^{}2[/tex][tex]\left|B\right|[/tex]

    where [tex]\left|M_{}11\right|[/tex] is the minor of a[tex]_{}11[/tex]

    I know what a symmetric tridiagonal matrix is.
    Is the minor oa a11 not just a11, the minor is the deterninant of a smaller part of a matrix right? but since a11 in only one entry is it not the minor as well?

    i'm not sure where to start this...
     
  2. jcsd
  3. Feb 8, 2009 #2
    A = [tex]
    \begin{pmatrix}a11 & a12 & 0 & 0 & ... \\ a21 & a22 & a23 & 0 & ... \\ 0 & a32 & a33 & a34 & 0 & ...\\ 0 & 0 & a43 & a44 & a45 & ... \end{pmatrix}

    [/tex]

    B = [tex]
    \begin{pmatrix}a33 & a34 & 0 & 0 & ... \\ a43 & a44 & a45 & 0 & ... \\ 0 & a54 & a55 & a56 & 0 & ...\\ 0 & 0 & a65 & a66 & a67 & ... \end{pmatrix}

    [/tex]

    B = [tex]
    \begin{pmatrix}a22 & a23 & 0 & 0 & ... \\ a32 & a33 & a34 & 0 & ... \\ 0 & a43 & a44 & a45 & 0 & ...\\ 0 & 0 & a54 & a55 & a56 & ... \end{pmatrix}

    [/tex]

    a12[tex]^{}2[/tex] = a12 x a21 because its symetric
     
  4. Feb 8, 2009 #3

    Mark44

    Staff: Mentor

    The minor of an entry in a matrix is the submatrix made up of all rows and columns that don't include that entry. For example, the minor [itex]M_11[/itex] of entry [itex]a_11[/itex] is the (n - 1) x (n - 1) matrix whose upper-left entry is [itex]a_2[/itex]. A minor is a matrix, and is different from its determinant.

    You're on the right track. Matrix B is as you show it in the first equation for B, with its upper-left entry of a33. I don't know what the other equation for B represents with its upper-left entry of a22.

    To evaluate |A| by minors, you'll get a11 * M11 - a12 * M12, where M12 is the submatrix of all entries not in row 1 and column 2. The 1st column of M12 has only one nonzero entry in it: a21 (which by symmetry = a12). When you expand A12, going down the first column, you'll get a21 * |B|. Be sure to include the appropriate signs associated with a12 and a21.

    Hope that helps
     
  5. May 14, 2009 #4
    I have a question on tridiagonal symmetric matrices..Is there a way to diagonalise it analytically like applying similarity transformations or in terms of Block matrices?..Thanks
     
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