Diagonal scaling to improve eigenvalue accuracy
[T,B] = balance(A)
[S,P,B] = balance(A)
B = balance(A)
B = balance(A,'noperm')
[T,B] = balance(A) returns a similarity transformation T such that B = T\A*T, and B has, as nearly as possible, approximately equal row and column norms. T is a permutation of a diagonal matrix whose elements are integer powers of two to prevent the introduction of roundoff error. If A is symmetric, then B == A and T is the identity matrix.
[S,P,B] = balance(A) returns the scaling vector S and the permutation vector P separately. The transformation T and balanced matrix B are obtained from A, S, and P by T(:,P) = diag(S) and B(P,P) = diag(1./S)*A*diag(S).
This example shows the basic idea. The matrix A has large elements in the upper right and small elements in the lower left. It is far from being symmetric.
A = [1 100 10000; .01 1 100; .0001 .01 1] A = 1.0e+04 * 0.0001 0.0100 1.0000 0.0000 0.0001 0.0100 0.0000 0.0000 0.0001
Balancing produces a diagonal matrix T with elements that are powers of two and a balanced matrix B that is closer to symmetric than A.
[T,B] = balance(A) T = 1.0e+03 * 2.0480 0 0 0 0.0320 0 0 0 0.0003 B = 1.0000 1.5625 1.2207 0.6400 1.0000 0.7813 0.8192 1.2800 1.0000
To see the effect on eigenvectors, first compute the eigenvectors of A, shown here as the columns of V.
[V,E] = eig(A); V V = 0.9999 -0.9999 -0.9999 0.0100 0.0059 + 0.0085i 0.0059 - 0.0085i 0.0001 0.0000 - 0.0001i 0.0000 + 0.0001i
Note that all three vectors have the first component the largest. This indicates V is badly conditioned; in fact cond(V) is 8.7766e+003. Next, look at the eigenvectors of B.
[V,E] = eig(B); V V = 0.6933 -0.6993 -0.6993 0.4437 0.2619 + 0.3825i 0.2619 - 0.3825i 0.5679 0.2376 - 0.4896i 0.2376 + 0.4896i
Now the eigenvectors are well behaved and cond(V) is 1.4421. The ill conditioning is concentrated in the scaling matrix; cond(T) is 8192.
This example is small and not really badly scaled, so the computed eigenvalues of A and B agree within roundoff error; balancing has little effect on the computed results.
Balancing can destroy the properties of certain matrices; use it with some care. If a matrix contains small elements that are due to roundoff error, balancing might scale them up to make them as significant as the other elements of the original matrix.
Nonsymmetric matrices can have poorly conditioned eigenvalues. Small perturbations in the matrix, such as roundoff errors, can lead to large perturbations in the eigenvalues. The condition number of the eigenvector matrix,
cond(V) = norm(V)*norm(inv(V))
[V,T] = eig(A)
relates the size of the matrix perturbation to the size of the eigenvalue perturbation. Note that the condition number of A itself is irrelevant to the eigenvalue problem.
Balancing is an attempt to concentrate any ill conditioning of the eigenvector matrix into a diagonal scaling. Balancing usually cannot turn a nonsymmetric matrix into a symmetric matrix; it only attempts to make the norm of each row equal to the norm of the corresponding column.