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I Linear algebra ( symmetric matrix)

  1. Jan 16, 2017 #1
    I am currently brushing on my linear algebra skills when i read this
    For any Matrix A
    1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out )
    2)(A + At)/2 is symmetric
    Now my question is , why should it be divided by 2? doesnt just A + At alone give a symmetric matrix
     
  2. jcsd
  3. Jan 16, 2017 #2

    fresh_42

    Staff: Mentor

    It goes like this ##\text{ [itex] A^t [/itex] }## or ##\text{ ## A^t ## }##.

    The relevant formulas are ##(A \cdot B)^t = B^t \cdot A^t \, , \, (A+B)^t = A^t + B^t ## and ##(A^t)^t=A##.
    You are correct, the factor ##\frac{1}{2}## isn't necessary here. It usually is taken when ##A## is written as ##A = \frac{1}{2}(A+A^t) + \frac{1}{2}(A-A^t)##, i.e. as a sum of a symmetric matrix ##B=B^t=\frac{1}{2}(A+A^t)## and a skew-symmetric matrix ##C=-C^t=\frac{1}{2}(A-A^t)##. Here it is needed to get back ##A##, instead of ##2A##.
     
  4. Jan 16, 2017 #3
    Thank you for the clarification.
     
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