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HomeMatlabWhat Is the Numerical Vary of a Matrix? – Nick Higham

What Is the Numerical Vary of a Matrix? – Nick Higham


The numerical vary of a matrix Ainmathbb{C}^{ntimes n}, also called the discipline of values, is the set of complicated numbers

W(A) = biggl{, displaystylefrac{z^*Az}{z^*z}: 0ne zinmathbb{C}^n ,biggr}.

The set W(A) is compact and convex (a nontrivial property proved by Toeplitz and Hausdorff), and it accommodates all of the eigenvalues of A. For regular matrices it’s the convex hull of the eigenvalues. For a Hermitian matrix, W(A) is a section of the true axis, whereas for a skew-Hermitian matrix it’s a section of the imaginary axis.

The next determine plots in blue the boundaries of the the numerical ranges of 9 16times 16 matrices, with the eigenvalues proven as black dots. They have been plotted utilizing the perform fv within the Matrix Computation Toolbox. The matrices are from Anymatrix.

fov_plots.jpg

A way for computing the boundary of the numerical vary relies on the statement that for any zinmathbb{C},

notag  z^*Az = underbrace{z^* frac{1}{2}(A+A^*) z}_{mathrm{phantom{1234}realphantom{1234}}}         + underbrace{z^* frac{1}{2}(A-A^*) z}_{mathrm{pure~imaginary}}        qquad (1)

This means that the true a part of z^*Az/z^*z lies between the biggest and smallest eigenvalues of the Hermitian matrix (A+A^*)/2, which outline a vertical strip wherein the numerical vary lies. Since W(mathrm{e}^{mathrm{i}theta} A) = mathrm{e}^{mathrm{i}theta} W(A), we are able to apply the identical reasoning to the rotated matrix A(theta) = mathrm{e}^{mathrm{i}theta} A, and taking a variety of thetain[0,pi] we get hold of an approximation the boundary of the numerical vary.

The amount

notag  omega(A) = max bigl{, mathrm{Re}, w : w in W(A) ,bigr}

known as the numerical abscissa, and by (1), omega(A) is the biggest eigenvalue of the Hermitian matrix (A+A^*)/2. The numerical abscissa determines the speed of development of |mathrm{e}^{tA}| for small optimistic t.

Related to the numerical vary is the numerical radius

notag      r(A) = max : w in W(A) ,.

Word that r(A^*) = r(A). Additionally, r(A) ge rho(A), the place rho is the spectral radius (the biggest absolute worth of any eigenvalue), since W(A) accommodates the eigenvalues of A.

The numerical radius differs by at most an element 2 from the 2-norm:

notag       frac{1}{2} |A|_2 le r(A) le |A|_2. qquad (2)

When A is regular, r(A) = |A|_2.

The numerical radius is a matrix norm, however not a constant norm (that’s, r(AB) le r(A)r(B) doesn’t maintain on the whole). Nevertheless, it’s it true that

notag     r(A^k) le r(A)^k, quad kge 1.

Combining this with with the decrease sure in (2) provides

notag  |A^k|_2 le 2 r(A)^k, quad k ge 1,

so if we all know r(A) then we are able to sure |A^k|_2 for all k.

The numerical radius may be characterised as the answer of an optimization downside over the Hermitian matrices:

notag      r(A) = - maxbiggl{, lambda_{min}biggl(     begin{bmatrix} Z!! & A  A^*!! & -Z end{bmatrix} biggr):                Z=Z^*inmathbb{C}^{ntimes n} ,biggr}.

Notes and References

For proofs of the outcomes given right here see Horn and Johnson (2013) or Horn and Johnson (1991), and see the latter for particulars of the algorithm for computing the numerical vary. See Benzi (2020) for a dialogue of functions of the numerical vary in numerical evaluation.

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