[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

575.0. "SQRT( matrix )" by COMET::ROBERTS (Dwayne Roberts) Wed Sep 10 1986 21:21

    
    Are there any good techniques for obtaining the square root of a
    given nth-order matrix?  For example, what's the square root of
    
                                    3  -1
                                    1   0

    Does every nth-order matrix have a square root?
    
    Can a matrix have more than one square root?
    
T.RTitleUserPersonal
Name
DateLines
575.1Example of 2 SQRT's:TAV02::NITSANNitsan Duvdevani, Digital IsraelThu Sep 11 1986 10:425
>    Can a matrix have more than one square root?

I don't really remember my Algebra, but consider a 1x1 matrix :-)

Nitsan.
575.2Lots of 'em26205::YARBROUGHThu Sep 11 1986 12:024
    Matrices can have lots of square roots - for example, the unit matrix
    of order N has 2^N square roots consisting of +and- 1's in each
    place in the diagonal. A better question is, does there exist an
    algorithm for finding all the square roots?
575.3Eigenthings might helpTOOK::APPELLOFCarl J. AppellofThu Sep 11 1986 13:0412
575.4CLT::GILBERTeager like a childThu Sep 11 1986 23:391
    What are the square- and cube-roots of 2x2 and 3x3 identity matrices?
575.5Possible geometric interpretationENGINE::ROTHFri Sep 12 1986 14:249
For the square root cases, the matrix representation of any geometric
transformaton which is an involution, for example any reflection with
respect to any line thru the origin in 2 dimensions, or any reflection
with respect to any plane thru the origin in 3 dimensions.

I have to think about the cube root a moment (if you allow complex
numbers this changes things).

- Jim
575.6More generally,ENGINE::ROTHFri Sep 12 1986 16:0711
    I think the square roots of the identity matrix for N dimensions
    would be any similarity transformation of an N dimensional matrix
    with +/- 1 on the main diagonal.

    For the cube root case it would be any similatrity transformation of
    a 2 or 3 dimensional matrix with cube roots of unity on the main
    diagonal.  For the reals in 3 space this could be restricted to
    any similarity transformation of any rotation by +/- 120 degrees
    about any axis...

    - Jim
575.7Some SqrtsCOMET::ROBERTSDwayne RobertsSat Sep 13 1986 23:0450
    
    For the square root of the 2nd-order identity matrix, besides the
    matrices 
    
     1   0      -1   0       1   0      -1   0
     0   1	 0   1       0  -1       0  -1
    
    there is also the infinite set of matrices of the form 
    
    0    x
    1/x  0

    where x is any non-zero real or complex number.

    ========================
    
    In .0, I posed the question of the square roots of the matrix
    
    			3  -1
    			1   0
    
    just as a tickler.  There are at least two very different square roots
    to the matrix.  They are 
    
    		2  -1
    		1  -1
    
    and
    
    1/sqrt(5)*	4  -1
    		1   1
    
    The second was arrived at using Newton's method; i.e.,
    
    	X    =  (0.5) * { X  + [ C * INV(X ) ] }
         n+1               n              n
    
    where C is the matrix
    
    3  -1
    1   0
    
    and INV is the inverse function.  I started with X  being the identity
                                                      1
    matrix.
    
    I'm surprised that Newton's method apparently works with matrices.
    Because the inverse of X is used, it can, of course, only find
    non-singular roots. 
     
575.8CLT::GILBERTeager like a childSun Sep 14 1986 01:3515
    For the square root of the 2nd-order identity matrix, besides the matrices 
    
     1   0      -1   0       1   0      -1   0
     0   1	 0   1       0  -1       0  -1

    The only other solutions are:

     a   b
     c  -a  , where a^2 + bc = 1.

    That is, the sum of the elements on the major diagonal (the trace) is 0.
    I suspect that other roots of identity matrices also have interesting
    relationships on the trace.

					- Gilbert
575.9ENGINE::ROTHTue Sep 16 1986 06:2324
    The trace and determinant are both invariant under similarity transform,
    so the criteria I gave should hold in general.

    In .8,

	a  b
	c -a   is the form of the general similarity transform of either

	-1  0	   1  0
	 0  1  or  0 -1.


    Books on group theory as applied to quantum mechanics (such as by
    Tinkham or Hammermesh) contain interesting discussions of the theory of
    reducing matrices to canonical form under certain symmetry constraints.
    I came across this once when studying a microwave network analysis
    problem involving symmetric waveguide junctions, and it was fascinating
    stuff, involving group theory, linear algebra, geometry, and circuit
    theory...

    So it may be worth looking into for more info on roots of matrices and
    related matters.

    - Jim