This may confuse people and questioning the concept of continuous. The proof given above is arguably in correct There are two types of continuity concerning eigenvalues: 1 each individual eigenvalue is a usual continuous function such a representation does exist on a real interval but may not exist on a complex domain , 2 eigenvalues are continuous as a whole in the topological sense a mapping from the matrix space with metric induced by a norm to unordered tuples, i. Whichever continuity is used in a proof of the Gerschgorin disk theorem, it should be justified that the sum of algebraic multiplicities of eigenvalues remains unchanged on each connected region. A proof using the argument principle of complex analysis requires no eigenvalue continuity of any kind. In this kind of problem, the error in the final result is usually of the same order of magnitude as the error in the initial data multiplied by the condition number of A.
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Topics Analytics Data Visualization The eigenvalues of a matrix are not easy to compute. It is remarkable, therefore, that with relatively simple mental arithmetic, you can obtain bounds for the eigenvalues of a matrix of any size. Although the theorem holds for matrices with complex values, this article only uses real-valued matrices. An example of Gershgorin discs is shown to the right. However, two mathematical theorems tells us quite a lot about the eigenvalues of this matrix, just by inspection.
First, because the matrix is real and symmetric, the Spectral Theorem tells us that all eigenvalues are real. Although the eigenvalues for this matrix are real, the Gershgorin discs are in the complex plane.
The discs are visualized in the graph at the top of this article. The true eigenvalues of the matrix are shown inside the discs. For this example, each disc contains an eigenvalue, but that is not true in general. What is true, however, is that disjoint unions of discs must contain as many eigenvalues as the number of discs in each disjoint region. Therefore they each contain an eigenvalue.
The union of the other two discs must contain two eigenvalues, but, in general, the eigenvalues can be anywhere in the union of the discs. The visualization shows that the eigenvalues for this matrix are all positive. That means that the matrix is not only symmetric but also positive definite. You can predict that fact from the Gershgorin discs because no disc intersects the negative X axis. A strictly diagonally dominant matrix is one for which the magnitude of each diagonal element exceeds the sum of the magnitudes of the other elements in the row.
Geometrically, this means that no Gershgorin disc intersects the origin, which implies that the matrix is nonsingular. So, by inspection, you can determine that his matrix is nonsingular. Gershgorin discs for correlation matrices The Gershgorin theorem is most useful when the diagonal elements are distinct. For repeated diagonal elements, it might not tell you much about the location of the eigenvalues. For example, all diagonal elements for a correlation matrix are 1.
Consequently, all Gershgorin discs are centered at 1, 0 in the complex plane. The discs and the actual eigenvalues of this matrix are shown in the following graph. Not only does the Gershgorin theorem bound the magnitude of the real part of the eigenvalues, but it is clear that the imaginary part cannot exceed 2. Conclusions In summary, the Gershgorin Disc Theorem provides a way to visualize the possible location of eigenvalues in the complex plane. You can use the theorem to provide bounds for the largest and smallest eigenvalues.
I was never taught this theorem in school. I learned it from a talented mathematical friend at SAS. I use this theorem to create examples of matrices that have particular properties, which can be very useful for developing and testing software.
The Gershgorin Disc Theorem shows the effect of ridging a matrix is to translate all of the Gershgorin discs to the right, which moves the eigenvalues away from zero while preserving their relative positions. You can download the SAS program that I used to create the images in this article. Further reading There are several papers on the internet about Gershgorin discs. It is a favorite topic for advanced undergraduate projects in mathematics.
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