Results 1 to 3 of 3

Math Help - The rows of this unitary matrix form an orthonormal set but the columns don't. Why?

  1. #1
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    The rows of this unitary matrix form an orthonormal set but the columns don't. Why?

    The rows of this ( {{1/3 - 2/3i,2/3i},{-2/3i,-1/3-2/3i}} - Wolfram|Alpha ) unitary matrix form an orthonormal set but, the columns of A do not form an orthonormal set despite this ( https://en.wikipedia.org/wiki/Unitar...ent_conditions ) Wikipedia article saying that they should.Could someone please help me understand why the columns of A do not form an orthonormal set for this unitary matrix?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,313
    Thanks
    695

    Re: The rows of this unitary matrix form an orthonormal set but the columns don't. Wh

    Hmm...let's see:

    Let's take the inner product of the first two columns:

    \left(\frac{1}{3} - \frac{2}{3}i\right)\left(\overline{\frac{2}{3}i} \right) + \left(-\frac{2}{3}i \right)\left(\overline{-\frac{1}{3} - \frac{2}{3}i} \right)

    =\left(\frac{1}{3} - \frac{2}{3}i\right)\left(-\frac{2}{3}i \right) + \left(-\frac{2}{3}i \right)\left(-\frac{1}{3} + \frac{2}{3}i \right)

    = -\frac{4}{9} - \frac{2}{9}i + \frac{4}{9} + \frac{2}{9}i = 0.

    So it appears the columns are orthogonal.

    Taking norms, for the first column we have:

    \sqrt{\left(\frac{1}{3} - \frac{2}{3}i \right)\left(\frac{1}{3} + \frac{2}{3}i \right) + \left(-\frac{2}{3}i \right)\left(\frac{2}{3}i \right)}

    = \sqrt{\frac{1}{9} + \frac{4}{9} + \frac{4}{9}} = \sqrt{1} = 1.

    I leave it to you to show that the second column is also a unit vector. I don't see the problem.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Re: The rows of this unitary matrix form an orthonormal set but the columns don't. Wh

    I had just figured out that I was doing the dot products incorrectly and was going to mention it ... sorry that you had to type that out but, thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 14th 2012, 07:57 AM
  2. Finding recessive rows and columns in a matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 21st 2011, 11:23 AM
  3. adding rows and columns....
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: August 29th 2010, 10:47 PM
  4. Figuring out rows and columns after matrix multiplication
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 23rd 2009, 05:18 AM
  5. multiplying columns and rows
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 8th 2009, 04:33 AM

Search Tags


/mathhelpforum @mathhelpforum