This is an example of a singular value decomposition. A = U E V^twith( LinearAlgebra ):
A := Matrix( [[ -2, 1, 2], [6, 6, 3]] );Compute the singular values, which are the square roots of the spectrum( A*A ). Note that A*A is hermitian,
so if A is real, all of the eigenvalues will be positive reals as well.( singVals, eVecs ) := Eigenvectors( HermitianTranspose( A ) . A ):
Map( x->sqrt(x), singVals ):
( cDim, rDim ) := Dimension( eVecs ):
for i from 1 to Dimension( singVals ) do
for j from i to Dimension( singVals ) do
if ( singVals[ i ] < singVals[ j ] ) then
tempVal := singVals[ i ];
singVals[ i ] := singVals[ j ];
singVals[ j ] := tempVal;
tempVec := Column( eVecs, i );
eVecs[ 1..rDim, i ] := Column( eVecs, j );
eVecs[ 1..rDim, j ] := tempVec;
end if;
end do;
end do;
( singVals, eVecs );
We now normalize all of the eigenvectors. Since A*A is hermitian, we know that all the eigenvectors are
already orthogonal. These normalized eigenvectors give us the matrix V.V := Matrix( eVecs ):
for i from 1 to cDim do
V[ 1..rDim, i ] := ( 1 / sqrt( Transpose( Column( V, i ) ) . Column( V, i ) ) ) . Column( V, i );
end do:
V;The E matrix is built from the nonzero singular values on the diagonal. The overall shape is the same as A.E := Matrix( Dimension( A ) ):
for i from 1 to Dimension( singVals ) do
if ( singVals[ i ] <> 0 ) then
E[ i, i ] := singVals[ i ];
end if:
end do:
E;Now I'm trying to build UU := ( A . Column( V, 1 ) ) / singVals[ 1 ]:
for i from 2 to Dimension( singVals ) do
if ( singVals[ i ] <> 0 ) then
U := < U | (( A . Column( V, i ) ) / singVals[ i ] ) >;
end if:
end do;
U;
That's it. A = U E V^tEqual( A, U . E . Transpose( V ) );All this work could easily have been achieved using Maple's built-in functionality.
Note that while the built in function can spit out the singular values, it doesn't build a prettySigma (E) matrix like I did above.( mU, mVt ) := SingularValues( A, output=['U', 'Vt'] );
Equal( A, mU . E . mVt );TTdSMApJNVJUQUJMRV9TQVZFLzIxNzAxNzE2WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjJyIjIiQhIiMiIiciIiJGKCIiIyIiJEYmCg==TTdSMApJM1JUQUJMRV9TQVZFLzY0MzgzMlgqJSphbGdlYnJhaWNHNiI2IltnbCEjJSEhISIkIiQiIioiIiQiIiFGJgo=TTdSMApJNVJUQUJMRV9TQVZFLzIxNzI0OTkyWCwlKmFsZ2VicmFpY0c2IjYiW2dsISIlISEhIyoiJCIkIiIjRiciIiIhIiIjRihGJ0YoRipGKUYoCkYmCg==TTdSMApJM1JUQUJMRV9TQVZFLzY0MzgzMlgqJSphbGdlYnJhaWNHNiI2IltnbCEjJSEhISIkIiQiIioiIiQiIiFGJgo=TTdSMApJNVJUQUJMRV9TQVZFLzIxNzI0OTkyWCwlKmFsZ2VicmFpY0c2IjYiW2dsISIlISEhIyoiJCIkIiIjRiciIiIhIiIjRihGJ0YoRipGKUYoCkYmCg==TTdSMApJNVJUQUJMRV9TQVZFLzIzMzAwNjQwWCwlKmFsZ2VicmFpY0c2IjYiW2dsISIlISEhIyoiJCIkIyIiIyIiJEYnIyIiIkYpIyEiI0YpRipGCidGKkYsRidGJgo=TTdSMApJNVJUQUJMRV9TQVZFLzI0NDYyOTcyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjJyIjIiQiIioiIiFGKCIiJEYoRihGJgo=TTdSMApJNFJUQUJMRV9TQVZFLzU0MDYxMTZYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMlIiMiIyIiISIiIkYoRidGJgo=TTdSMApJNVJUQUJMRV9TQVZFLzI0ODMwNjIwWCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjJSIjIiMwMDAwMDAwMDAwMDAwMDAwM0ZGMDAwMDAwCjAwMDAwMDAzRkYwMDAwMDAwMDAwMDAwMDAwMDAwMDAwMDAwMDAwMEYmCg==TTdSMApJNFJUQUJMRV9TQVZFLzQ1NjM3MjhYLCUpYW55dGhpbmdHNiI2IltnbCciJSEhISMqIiQiJDNGRTU1NTU1NTU1NTU1NTVCRkU1NTU1NTU1CjU1NTU1NjNGRDU1NTU1NTU1NTU1NTUzRkU1NTU1NTU1NTU1NTU0M0ZENTU1NTU1NTU1NTU1NkJGRTU1NTU1NTU1NTU1NTUzRkQ1NTU1NTUKNTU1NTU1NTNGRTU1NTU1NTU1NTU1NTYzRkU1NTU1NTU1NTU1NTU1RiYK