Math 439 Homework 1 - Fall 2008

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    HOMEWORK #1 due 9-16

    NOTE: In the following, _ means subscript and ^ means superscript. The expression Sum(i=a,b) h(i) (for example) means the same as Sumi=1b h(i), which is the sum of h(i) for i=a,a+1,...,b. Also, ``ne'' means ``not equal to'', so that x ne 0 means that x is not equal to zero.

    1. Consider the matrix 

                    (  -2    9   -7    3  -2  )
 A = (a_{ij}) = (   5   13   14    6   0  )
                (  11    0   17   -2  -3  )
    Calculate
    (i)   a_{2+} = Sum(j=1,5) a_{2j}   (Hint: You should get 38)
    (ii)   Sum(i=1,3) a_{i4}
    (iii)   Sum(i=1,3) a_{ii}
    (iv)   Sum(u=1,3) a_{u2}a_{u5}

    2. Write down the three 3 by 3 matrices A=(a_{ij}) with entries

    (i) a_{ij}=i+j-2,
    (ii) a_{ij}=i(j-1)    and
    (iii) a_{ij}=j-i
    where i,j=1,2,3.

    3. Let B be the matrix 

             B = (  1  7 )
             (  4  3 )
             ( -3  6 )
    (i) Find the matrices BB' and B'B
    (ii) Find tr(BB') and tr(B'B) (where tr means trace)

    4. Consider the matrix 

     A = (  1   3   7 )
     (  2   6  14 )
     ( -1  -3  -7 )
    Find the matrices A2 and A'A.  Are they the same? Are you sure?

    5. Let A be the matrix 

     A = ( -1   1  -1 )
     (  0   2   1 )
     (  1   1  -1 )
    (i) Note that the columns of A are mutually orthogonal. Normalize the columns of A by dividing each column by its length. Denote the resulting matrix by C.
    (ii) Show that C is an orthogonal matrix, and that CC'=C'C=I_3.

    6. Let A be an nxn matrix and j a column vector of 1s.

    (i) Show that j'A is a row vector whose elements are the column sums of A.
    (ii) Show that Aj is a column vector whose elements are the row sums of A.
    (Hint: Express all matrix products as sums over matrix entries.)

    7. Let A=xy' be the outer product of two 3x1 column vectors x and y. (Since A is the product of a 3x1 matrix x and a 1x3 matrix y', A is 3x3. In fact, A_{ij}=x_iy_j for i,j=1,2,3.) In contrast, the inner product x'y is the product of a 1x3 matrix x' and a 3x1 matrix y, so that x'y is 1x1, which is the number x'y=Sum(i=1,3) x_iy_i. Show that

    (i)   If y'x=1, then A2=A
    (ii)   If y'x=0, then A2=0
    (iii)   tr(xy')=x'y.   (Warning: This is not a misprint:   xy' and x'y are different objects.)
    (Hint: Carry out the indicated matrix multiplications, or write out expressions in terms of sums over indices.)

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