| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 575.1 | Example of 2 SQRT's: | TAV02::NITSAN | Nitsan Duvdevani, Digital Israel | Thu Sep 11 1986 07:42 | 5 |
| > Can a matrix have more than one square root?
I don't really remember my Algebra, but consider a 1x1 matrix :-)
Nitsan.
|
| 575.2 | Lots of 'em | 26205::YARBROUGH | | Thu Sep 11 1986 09:02 | 4 |
| 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?
|
| 575.3 | Eigenthings might help | TOOK::APPELLOF | Carl J. Appellof | Thu Sep 11 1986 10:04 | 12 |
| Dredging up my linear algebra, I'd start with finding the eigenvectors
and eigenvalues of the matrix A (only square matrices qualify, of
course). Take the square root of each of the eigenvalues and form
a diagonal matrix of those numbers (maybe �, I haven't checked).
Multply this on the right by the matrix of eigenvalues to get a
matrix B such that
T
B B = A.
Now, if you want BB = A, that's another kettle of fish.
|
| 575.4 | | CLT::GILBERT | eager like a child | Thu Sep 11 1986 20:39 | 1 |
| What are the square- and cube-roots of 2x2 and 3x3 identity matrices?
|
| 575.5 | Possible geometric interpretation | ENGINE::ROTH | | Fri Sep 12 1986 11:24 | 9 |
| 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
|
| 575.6 | More generally, | ENGINE::ROTH | | Fri Sep 12 1986 13:07 | 11 |
| 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
|
| 575.7 | Some Sqrts | COMET::ROBERTS | Dwayne Roberts | Sat Sep 13 1986 20:04 | 50 |
|
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.
|
| 575.8 | | CLT::GILBERT | eager like a child | Sat Sep 13 1986 22:35 | 15 |
| 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
|
| 575.9 | | ENGINE::ROTH | | Tue Sep 16 1986 03:23 | 24 |
| 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
|