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Is the 0 matrix upper triangular?

  1. Oct 27, 2012 #1
    Is the 0 matrix upper triangular?

    The reason I ask is because I'm trying to determine whether something is a subspace.
    The problem is determine whether the subset S of M2x2 is a subspace where S is the set of all upper triangular matrices.


    So these 3 must be satisfied:
    1) 0 vector(matrix) is in S
    2) if U and V are in S then U+V is in S
    3) if V is in S, then cV where c is a scalar is in S


    if 0 matrix is in S then that means
    S=
    0 0
    0 0

    But is that still upper triangular?
    Upper triangular is defined as having all entries below the main diagnol be 0. I thought a main diagonal was having a nonzero # along the diagonal?
     
  2. jcsd
  3. Oct 27, 2012 #2

    Mark44

    Staff: Mentor

    The main diagonal consists of the entries from the upper left corner to the lower right corner. There's nothing in this definition that requires any particular values.

    So, yes, the zero matrix is upper triangular (and lower triangular, too).
     
  4. Oct 27, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Saying that certain number must be 0 doesn't mean that other cannot also be 0.
     
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