Results 1 to 8 of 8

Math Help - Quaternion

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Quaternion

    Let q=a+bi+cj+dk be a nonzero quaternion.
    (a) Prove that the left multiplication L_q:\mathbb{H}\to\mathbb{H} given by L_q(x)=qx is an isomorphism.

    L_q(x_1)=L_q(x_2)\Rightarrow qx_1=qx_2
    By def., the Quaternion Algebra is a vector space equipped with a ring. Therefore, q^{-1} exist.
    q^{-1}qx_1=q^{-1}qx_2\Rightarrow x_1=x_2
    Thus, L_q(x) is monic.

    y=L_q(x)=qx\Rightarrow q^{-1}y=x
    L_q(q^{-1}y)=qq^{-1}y=y
    Therefore, f is an epimorphism and f is an isomorphism.

    (b) Find the matrix of L_q relative to the basis 1,i,j,k\in\mathbb{H}

    Can someone walk step by step through this piece? I don't remember what to do.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,279
    Thanks
    671

    Re: Quaternion

    vector spaces equipped with a ring do not necessarily have inverses. for example, what is x^-1 in R[x]?

    however, quaternions are special, they form a division ring, which is what we need for (multiplicative) inverses to exist.

    it is this same property (existence of inverses) that allows you to prove surjectivity.

    but those are minor points, easily fixed by the addition of a single word ("divison" in front of ring) in your proof.

    you also need to show that L_q is a linear map. i recommend invoking the distributive law, and the fact that real numbers

    commute with quaternions.

    for part (b) you are looking for a 4x4 (real) matrix that corresponds to (left) mulitplication by q. fix an arbitrary quaternion q' in H,

    and calculate qq', collecting terms by basis elements. for example, if q' = a' + b'i + c'j +d'k, the real term will be:

    aa' - bb' - cc' - dd', so the first row should be [a -b -c -d].
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Quaternion

    Quote Originally Posted by Deveno View Post
    aa' - bb' - cc' - dd', so the first row should be [a -b -c -d].
    Why is the first row a -b -c -d and not aa' -bb' -cc' dd'?

    \begin{bmatrix}aa'&-bb'&-cc'&-dd'\\ab'&a'b&cd'&-dc'\\ac'&a'c&-bd'&db'\\ad'&a'd&bc'&-cb'\end{bmatrix}

    This is what I have obtained.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,279
    Thanks
    671

    Re: Quaternion

    if q' = a' + b'i + c'j + d'k, in the basis {1,i,j,k} q' has coordinates (a',b',c',d'), so your matrix multiplication should look like this:

    \begin{bmatrix}a&-b&-c&-d\\b&a&d&-c\\c&d&a&-b\\d&-c&b&a \end{bmatrix} \begin{bmatrix}a'\\b'\\c'\\d' \end{bmatrix} = \begin{bmatrix}aa'-bb'-cc'-dd'\\ab'+a'b+cd'-c'd\\ac'+a'c-bd'+b'd\\ad'+a'd+bc'-b'c \end{bmatrix}

    the 4x4 matrix is the matrix for L_q, the result is qq' (as a quaternial product).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Quaternion

    Quote Originally Posted by Deveno View Post
    if q' = a' + b'i + c'j + d'k, in the basis {1,i,j,k} q' has coordinates (a',b',c',d'), so your matrix multiplication should look like this:

    \begin{bmatrix}a&-b&-c&-d\\b&a&d&-c\\c&-d&a&b\\d&c&-b&a \end{bmatrix} \begin{bmatrix}a'\\b'\\c'\\d' \end{bmatrix} = \begin{bmatrix}aa'-bb'-cc'-dd'\\ab'+a'b+cd'-c'd\\ac'+a'c-bd'+b'd\\ad'+a'd+bc'-b'c \end{bmatrix}

    the 4x4 matrix is the matrix for L_q, the result is qq' (as a quaternial product).
    I don't understand the first matrix. Why is it that?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,279
    Thanks
    671

    Re: Quaternion

    um, there were some errors in it. i hope they're fixed now.

    look, L_q is a linear transformation from H to H. we are regarding H as a 4-dimensional real vector space. what does a matrix from R^4 to R^4 look like?

    we are looking for a 4x4 matrix L_q such that L_q(q') = qq'.

    that is, this matrix should take q'-->qq'. what kind of thing takes a 4-vector to a 4-vector?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Quaternion

    Quote Originally Posted by Deveno View Post
    um, there were some errors in it. i hope they're fixed now.

    look, L_q is a linear transformation from H to H. we are regarding H as a 4-dimensional real vector space. what does a matrix from R^4 to R^4 look like?
    I still don't understand how you came up with that matrix.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,279
    Thanks
    671

    Re: Quaternion

    if q = a + bi + cj + dk and q' = a' + b'i + c'j + d'k, then qq' = (aa' - bb' - cc' - dd') + (ab' + a'b + cd' - c'd)i + (ac' + a'c - bd' + b'd)j + (ad' + a'd + bc' - b'c)k, right?

    in the basis {1,i,j,k} this is the 4-vector (aa'-bb'-cc'-dd',ab'+a'b+cd'-c'd,ac'+a'c-b'd+b'd,ad'+a'd+bc'-b'c).

    now, the first entry of that 4-vector is the inner product of the first row-vector of the matrix for Lq with (a',b',c',d').

    the only part of the real number aa'-bb'-cc'-dd' involving a' is the term aa', hence the first entry of the first row for Lq is a.

    similarly, the only term involving b' is the term -bb', so b' must get multplied by -b, c' must get multiplied by -c, d' must get multiplied by -d.

    so the first row for Lq is [a -b -c -d].

    to find the second row for Lq, we look at the "i" coefficient in the product qq'. what is a' multiplied by in the i coefficient? well the only a' term in the i coefficient is ab' (=ba', since these are real numbers).

    so the first entry of the 2nd row for Lq is b. note that each "primed" entry of q' appears just once in every coefficient of qq'. so the "other part" of the term

    involving that coefficient of q' (whether it is a',b',c' or d') must be the entry for Lq in that row of Lq. it's just "fill in the blanks" using matrix multiplication rules.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Is this the Quaternion group?
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: December 1st 2011, 06:23 AM
  2. [SOLVED] quaternion algebra
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: August 29th 2011, 10:02 AM
  3. Finding Inverse of a Quaternion
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 17th 2011, 08:48 PM
  4. Quaternion Group
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: November 4th 2009, 10:26 AM
  5. Factor groups for the quaternion group?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 9th 2009, 01:29 AM

Search Tags


/mathhelpforum @mathhelpforum