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Math Help - Some matrix questions

  1. #1
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    Some matrix questions

    Hi!

    If P and Q are n x n stochastic matrices, why is PQ a stochastic matrix?
    And why is P^k a stochastic matrix (induction proof)?

    Find the basis for null space to the following matrix:
    0 0,7 0 0 0
    0 -1 0,6 0 0
    0 0,3 -1 0,65 0
    0 0 0,4 -1 0
    0 0 0 0,35 0

    (I'm sorry but I haven't learned LaTeX, and I'm so busy right now...)

    I suppose you have to row reduce, but I'm quite new to this, so I have problem with row reducing this one...

    Help is highly appreciated!
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  2. #2
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    Re: Some matrix questions

    Quote Originally Posted by expresstrain View Post
    Hi!

    If P and Q are n x n stochastic matrices, why is PQ a stochastic matrix?
    And why is P^k a stochastic matrix (induction proof)?

    Find the basis for null space to the following matrix:
    0 0,7 0 0 0
    0 -1 0,6 0 0
    0 0,3 -1 0,65 0
    0 0 0,4 -1 0
    0 0 0 0,35 0

    (I'm sorry but I haven't learned LaTeX, and I'm so busy right now...)

    I suppose you have to row reduce, but I'm quite new to this, so I have problem with row reducing this one...

    Help is highly appreciated!
    What have you tried?
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  3. #3
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    Re: Some matrix questions

    Your matrix, if I read this correctly is
    \begin{bmatrix}0 & 0,7 & 0 & 0 & 0\\ 0 & -1 & 0,6 & 0 & 0 \\ 0 & 0,3 & -1 & 0,65 & 0 \\ 0 & 0,4 & -1 & 0 \\ 0 & 0 & 0 & 0,35 & 0\end{bmatrix}

    While "row reduction" will often simplify the calculations, if you are not comfortanble with it yet you don't have to use it.
    The "null space" for this matrix is defined as the set of vectors \begin{bmatrix}u \\ v\\ x \\ y \\ z\end{bmatrix} such that
    \begin{bmatrix}0 & 0,7 & 0 & 0 & 0\\ 0 & -1 & 0,6 & 0 & 0 \\ 0 & 0,3 & -1 & 0,65 & 0 \\ 0 & 0,4 & -1 & 0 \\ 0 & 0 & 0 & 0,35 & 0\end{bmatrix}\begin{bmatrix}u \\ v\\ x \\ y \\ z\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}.

    That gives the 5 equations 0,7v= 0; -v+ 0,6x= 0; 0,3v- x+ 0,65y= 0; 0,4x- y= 0, and 0,35y= 0. Those should be easy to solve. An important point for this problem is that the first and last columns are all "0"s so there is no "u" or "z" in any of those equations.
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  4. #4
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    Re: Some matrix questions

    Ok, thanks a lot! But I'm still an idiot on this topic:

    You got 5 equations, and everyone of them is 0 ?? v=0, x=0 and y=0 ?

    Does that mean that the basis of the null space is (0, 0, 0) ?
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  5. #5
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    Re: Some matrix questions

    no, it means we have 5 linear combinations of u,v,x,y and z that are all equal to 0.
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  6. #6
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    Re: Some matrix questions

    Ok, I'm a beginner on this, so: what is the next I should do?
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  7. #7
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    Re: Some matrix questions

    First, a "basis" must be a set of independent vectors and so cannot contain the 0 vector. Second, vectors in this space have five components, not three so (0, 0, 0) is not even in the space.

    The equations I derived from the definition of "Null Space" were 0,7v= 0; -v+ 0,6x= 0; 0,3v- x+ 0,65y= 0; 0,4x- y= 0, and 0,35y= 0. The first gives v= 0, obviously. With that, the second gives x= 0, and then y= 0. But remember that there are 5 components. v, x, and y must be 0 but, as I said before, there were no "u" or "z" in those equations so u and z can be any numbers. A vector in the null space must be of the form <u, 0, 0, 0, z>= <u, 0, 0, 0, 0>+ <0, 0, 0, 0, z>= u<1, 0, 0, 0, 0>+ z<0, 0, 0, 0, 1>.
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