What Is the Inertia of a Matrix? – Nick Higham

The inertia of an actual symmetric matrix is a triple, written , the place is the variety of optimistic eigenvalues of , is the variety of unfavourable eigenvalues of , and is the variety of zero eigenvalues of .

The rank of is . The distinction is named the signature.

On the whole it isn’t attainable to find out the inertia by inspection, however some deductions could be made. If has each optimistic and unfavourable diagonal components then and . However on the whole the diagonal components don’t inform us a lot in regards to the inertia. For instance, here’s a matrix that has optimistic diagonal components however just one optimistic eigenvalue (and this instance works for any ):

>> n = 4; A = -eye(n) + 2*ones(n), eigA = eig(sym(A))'
A =
     1     2     2     2
     2     1     2     2
     2     2     1     2
     2     2     2     1
eigA =
[-1, -1, -1, 7]

A congruence transformation of a symmetric matrix is a change for a nonsingular matrix . The result’s clearly symmetric. Sylvester’s regulation of inertia (1852) says that the inertia is preserved underneath congruence transformations.

Theorem 1 (Sylvester’s regulation of inertia).

If is symmetric and is nonsingular then .

Sylvester’s regulation provides a option to decide the inertia with out computing eigenvalues: discover a congruence transformation that transforms to a matrix whose inertia could be simply decided. A factorization does the job, the place is a permutation matrix, is unit decrease triangular, and is diagonal Then , and could be learn off the diagonal of . This factorization doesn’t all the time exist, and if it does exist is could be numerically unstable. A block factorization, through which is block diagonal with diagonal blocks of measurement or , all the time exists, and its computation is numerically steady with an appropriate pivoting technique corresponding to symmetric rook pivoting.

For the matrix above we will compute a block factorization utilizing the MATLAB ldl perform:

>> [L,D,P] = ldl(A); D
D =
   1.0000e+00   2.0000e+00            0            0
   2.0000e+00   1.0000e+00            0            0
            0            0  -1.6667e+00            0
            0            0            0  -1.4000e+00

For the reason that main 2-by-2 block of has unfavourable determinant and therefore one optimistic eigenvalue and one unfavourable eigenvalue, it follows that has one optimistic eigenvalue and three unfavourable eigenvalues.

A congruence transformation preserves the indicators of the eigenvalues however not their magnitude. A results of Ostrowski (1959) bounds the ratios of the eigenvalues of the unique and reworked matrices. Let the eigenvalues of a symmetric matrix be ordered .

Theorem (Ostrowski).

For a symmetric and ,

the place .

The concept reveals that the additional is from being orthogonal the larger the potential change within the eigenvalues.

Lastly, we notice that the whole lot right here generalizes to complicated Hermitian matrices by changing transpose by conjugate transpose.