The Gershgorin circle theorem lets you bound where the eigenvalues of a matrix must be from its entries alone, without ever computing them. Move the diagonal and off-diagonal entries of a 3x3 symmetric matrix and watch every eigenvalue stay trapped inside the discs drawn in the complex plane.
Parameters
Diagonal entry a11
Diagonal entry of row 1. It becomes the centre of disc 1
Diagonal entry a22
Diagonal entry of row 2. It becomes the centre of disc 2
Diagonal entry a33
Diagonal entry of row 3. It becomes the centre of disc 3
Off-diagonal magnitude offMag
Shared value of every off-diagonal entry. Each disc radius equals 2*offMag
Results
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Disc 1 (centre, radius)
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Disc 2 (centre, radius)
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Disc 3 (centre, radius)
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Largest eigenvalue (actual)
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Smallest eigenvalue (actual)
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Gershgorin upper bound
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Complex plane — Gershgorin discs and eigenvalues
The horizontal axis is the real axis and the vertical axis is the imaginary axis. The three translucent discs are the row discs; the dots on the real axis are the actual eigenvalues. Every dot lies inside at least one disc.
Discs and eigenvalues (1-D layout on the real axis)
The Gershgorin disc for row i, centred on the diagonal entry a_ii with a radius R_i equal to the sum of the absolute off-diagonal entries in that row. Every eigenvalue lies in the union of the discs, and a disc disjoint from all others contains exactly one eigenvalue.
The interval on the real axis that contains the whole spectrum. The upper bound caps the largest eigenvalue and the lower bound floors the smallest one.
What is the Gershgorin Circle Theorem?
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"Gershgorin circle theorem" is an unfamiliar name to me. What does this theorem actually do?
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Roughly speaking, it tells you "roughly where the eigenvalues are, without computing them". Computing eigenvalues exactly becomes fairly expensive as the matrix grows. But with the Gershgorin theorem you can look at the numbers in the matrix and declare "the eigenvalues can only be inside this region". It is one of the few genuine "free lunches" in numerical linear algebra.
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Knowing the range without computing sounds a bit like magic. How is the range determined?
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You draw one circle in the complex plane for each row of the matrix. The centre is that row's diagonal entry. The radius is the sum of the absolute values of all the other numbers in that row. That is the Gershgorin disc. The theorem guarantees that every eigenvalue lies inside the union of these discs. Move the a11 slider on the left and you will see disc 1 slide left and right along the real axis.
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I see! So what happens to the discs when I raise the off-diagonal magnitude (offMag)?
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Good observation. As offMag grows, the sum of the absolute off-diagonal entries in each row grows, so the disc radii get bigger and bigger. The 3x3 matrix in this tool has two off-diagonal entries per row, so the radius is uniformly 2*offMag. Big radii make the discs overlap and the eigenvalues spread away from the diagonal entries. Conversely, set offMag to 0 and the discs collapse to radius-zero points, and the eigenvalues become exactly the diagonal entries.
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I found a state where one disc sits alone, separated from the others. Does that mean anything?
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It means a lot. There is a second Gershgorin theorem that says "if a disc does not overlap any other disc, that disc contains exactly one eigenvalue". So when you spot an isolated disc, you can assert that there is exactly one eigenvalue right next to that diagonal entry. That is strong information — it tells you the count of eigenvalues, not just a region.
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In practice, when do engineers use this theorem?
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A common use is checking the convergence of iterative methods. Whether the Jacobi or Gauss-Seidel method converges depends on whether the spectral radius (the largest absolute eigenvalue) of a certain matrix is below 1. If Gershgorin tells you "all eigenvalues are inside these discs", you get an upper bound on the spectral radius for free. It is also used to confirm a matrix is positive-definite and to judge the stability of control systems. As a tool to "get a quick read" before running an eigenvalue solver, it is very handy.
Frequently Asked Questions
It states that every eigenvalue of an n by n matrix lies inside the union of n Gershgorin discs, one defined by each row. The disc for row i is centred on the diagonal entry a_ii and has a radius R_i equal to the sum of the absolute values of the off-diagonal entries in that row. The huge advantage is that you can bound where the eigenvalues must be from the matrix entries alone, without ever computing the eigenvalues themselves.
The radius R_i of the disc for row i is the sum of the absolute values of all entries in that row except the diagonal, R_i = sum over j not equal to i of |a_ij|. In this simulator every off-diagonal entry of the 3x3 matrix shares the same magnitude offMag, so every row has two off-diagonal entries and every radius equals R = 2*offMag. The larger the off-diagonal magnitude, the wider the discs, and the more the eigenvalues spread away from the diagonal entries.
Computing eigenvalues is generally an expensive task, but the Gershgorin theorem bounds their location instantly from the matrix entries. In practice it is used to estimate the spectral radius that governs the convergence of iterative methods, to confirm that a matrix is non-singular or positive-definite, to bound the stability of a system, and to test diagonal dominance. It also serves as a cheap sanity check before running a full eigenvalue solver.
By the second Gershgorin theorem, if the union of k discs is disjoint from the remaining discs, that union contains exactly k eigenvalues. In particular, if a single disc does not overlap any other disc, it contains exactly one eigenvalue. The smaller the off-diagonal magnitude, the more easily the discs separate, and the eigenvalues become localised one per disc near the diagonal entries.
Real-World Applications
Convergence of iterative solvers: When a linear system is solved with the Jacobi or Gauss-Seidel method, convergence depends on whether the spectral radius of the iteration matrix is below 1. Finding the spectral radius exactly needs an eigenvalue computation, but the Gershgorin theorem gives an upper bound immediately from the largest disc radius. If a matrix is diagonally dominant (each diagonal entry larger in magnitude than the sum of the absolute off-diagonal entries in its row), every disc avoids the origin, and the theorem also shows that convergence is guaranteed.
Estimating natural frequencies in structural and vibration analysis: In the generalised eigenvalue problem of the stiffness and mass matrices from a finite-element model, the eigenvalues correspond to the squares of the structure's natural frequencies. Before running a detailed eigenvalue extraction, knowing the location of the eigenvalues from the Gershgorin discs helps set the search interval for mode extraction and check whether the computed results are physically reasonable.
Stability assessment of control systems: For the linear system x-dot = Ax to be stable, every eigenvalue of A must have a negative real part. If all Gershgorin discs sit in the left half of the complex plane (the region of negative real part), you can conclude stability without computing the eigenvalues. In robust control, the discs are also used to assess the stability margin when the matrix entries carry uncertainty.
Sanity check for eigenvalue solvers: If the result returned by a numerical eigenvalue solver such as the QR algorithm falls outside the Gershgorin discs, it is a sign of a bug or numerical error. If the result lies correctly inside the discs, it provides an independent verification that supports the reliability of the answer. The larger the computation, the more valuable such cheap, reliable checks become.
Common Misconceptions and Pitfalls
A frequent misconception is assuming that exactly one eigenvalue sits inside each disc. This holds only when the discs are disjoint and do not overlap. Once discs overlap and merge into a single connected region, all you can say is that the connected region contains as many eigenvalues as the number of discs that formed it. For example, when three discs all overlap into one large region, you know there are three eigenvalues inside it, but not which eigenvalue belongs to which disc. As you raise offMag in this tool, the discs overlap one after another and this "exactly one per disc" property breaks down before your eyes.
Next, assuming the Gershgorin discs are the best possible estimate of the eigenvalues. The theorem is convenient and reliable, but the bound it gives is often loose (wider than the actual spectrum). A similarity transform of the matrix (for example scaling by a diagonal matrix) leaves the eigenvalues unchanged while reshaping the discs, so a tighter estimate can sometimes be obtained. You can also use column discs (the discs of the transpose) alongside the row discs, and the intersection of the two gives a stricter estimate. Gershgorin is a "fast first approximation", not the final word.
Finally, do not mistakenly assume that the discs of a symmetric matrix spread in the imaginary direction. For a general matrix the Gershgorin discs are genuine "circles" in the complex plane and the eigenvalues may be complex. But for the symmetric matrix this tool handles (a real symmetric matrix), the eigenvalues are theoretically guaranteed to be real. So the eigenvalues always line up on the real axis, inside the interval where the discs intersect the real axis. The eigenvalue dots on the canvas always lie on the real axis because the matrix is fixed to be symmetric. With a non-symmetric matrix, the eigenvalues leave the real axis and scatter across the complex plane.
How to Use
Enter diagonal elements (a11, a22, a33) representing matrix diagonal entries in your system matrix.
Set off-diagonal magnitude to define absolute values of off-diagonal elements; the simulator constructs a symmetric matrix and scales these perturbations.
Click Simulate to compute three Gershgorin discs centered at each diagonal element with radius equal to the sum of absolute off-diagonal entries in that row.
Compare actual eigenvalues (computed via QR decomposition) against the union of three discs to verify the theorem's containment guarantee.
Worked Example
Consider a 3x3 structural stiffness matrix with a11=850 MPa, a22=920 MPa, a33=780 MPa, and off-diagonal magnitude=150 MPa. Disc 1 centers at 850 with radius 300 (two off-diagonals per row). Disc 2 centers at 920 with radius 300. Disc 3 centers at 780 with radius 300. Computed eigenvalues might be 1180, 850, 350 MPa—all strictly within their corresponding discs. The Gershgorin upper bound on the largest eigenvalue is 920+300=1220 MPa, safely bracketing the actual maximum of 1180 MPa.
Practical Notes
Gershgorin bounds are often loose for weakly diagonally dominant matrices; increase diagonal dominance (larger a_ii relative to off-diagonals) to tighten predictions in finite-element assemblies.
Use this simulator to quickly check eigenvalue locations before expensive iterative solvers (Arnoldi, Lanczos) on large industrial FEA systems.
Off-diagonal symmetry is enforced; asymmetric matrices require separate analysis and may have complex eigenvalues outside real Gershgorin discs.