Square matrix
Matrix Size:
3
Gershgorin Circle Theorem Visualizer
Draw row or column Gershgorin discs and see where a matrix's eigenvalues must lie.
Results are calculated automatically as you enter data.
Result
| Disc | Center | Radius | Interval on real axis |
|---|
Formula and interpretation
▼ See explanations and tips below ▼
Related Calculators
What Is the Gershgorin Circle Theorem?
The Gershgorin circle theorem is a way to locate the eigenvalues of a square matrix without solving the full eigenvalue problem first. Instead of computing every eigenvalue exactly, it builds simple regions in the complex plane where the eigenvalues must be.
For each row or column of a matrix, the theorem creates a disc. The center of the disc comes from a diagonal entry of the matrix. The radius comes from the absolute values of the off-diagonal entries in the same row or column. The important conclusion is that every eigenvalue of the matrix lies inside at least one of these discs.
This is useful because eigenvalues can be difficult to compute by hand, especially for matrices larger than \(2 \times 2\). Gershgorin discs give a quick visual and numerical bound before doing a more detailed calculation.
Although the theorem is often called a “circle” theorem, the useful region is the filled-in disc, not just the boundary circle. If an eigenvalue lies in a Gershgorin disc, it may be anywhere inside that disc.
Why Gershgorin Discs Matter
Gershgorin discs are useful because they connect matrix entries directly to possible eigenvalue locations. You can often learn something important from the matrix before computing a characteristic polynomial or running a numerical eigenvalue algorithm.
Students use Gershgorin discs to understand how diagonal and off-diagonal entries affect eigenvalues. Instructors use them to show why diagonal dominance matters. Engineers and applied mathematicians use them as quick bounds when studying stability, iterative methods, or matrices that are close to diagonal.
The theorem is especially helpful when you need to answer questions such as:
- Are all eigenvalues likely to be far from zero?
- Can all eigenvalues be guaranteed to lie in the left half-plane?
- Are the eigenvalues confined to small regions near the diagonal entries?
- Do separated groups of discs guarantee separated groups of eigenvalues?
- Does the matrix look diagonally dominant enough to give useful bounds?
Gershgorin discs are not a replacement for exact eigenvalue computation, but they are a powerful first check.
Key Terms to Know
- Square matrix: A matrix with the same number of rows and columns. Eigenvalues are defined for square matrices.
- Diagonal entry: An entry of the form \(a_{ii}\), running from the upper-left to the lower-right of the matrix.
- Off-diagonal entry: Any entry \(a_{ij}\) where \(i \ne j\).
- Eigenvalue: A scalar \(\lambda\) such that \(A\mathbf{x} = \lambda \mathbf{x}\) for some nonzero vector \(\mathbf{x}\).
- Complex plane: The plane where a number \(z = x + yi\) is plotted with real part \(x\) and imaginary part \(y\).
- Disc center: The diagonal entry \(a_{ii}\) used as the center of a Gershgorin disc.
- Disc radius: The sum of absolute values of the off-diagonal entries in the selected row or column.
- Cluster: A connected group of overlapping Gershgorin discs.
- Algebraic multiplicity: The number of times an eigenvalue is counted as a root of the characteristic polynomial.
- Diagonal dominance: A situation where diagonal entries are large compared with the off-diagonal entries in their rows or columns.
How the Gershgorin Circle Theorem Works
Let \(A\) be an \(n \times n\) square matrix with entries \(a_{ij}\). For row mode, the \(i\)th Gershgorin radius is the sum of the absolute values of the off-diagonal entries in row \(i\):
$$
R_i^{\text{row}} = \sum_{j \ne i} |a_{ij}|
$$
The corresponding Gershgorin disc is
$$
D_i^{\text{row}} =
\left\{
z \in \mathbb{C} :
|z - a_{ii}| \le R_i^{\text{row}}
\right\}.
$$
The theorem says that every eigenvalue \(\lambda\) of \(A\) lies in at least one of these row discs:
$$
\lambda \in D_1^{\text{row}} \cup D_2^{\text{row}} \cup \cdots \cup D_n^{\text{row}}.
$$
Column mode uses the same diagonal centers, but it calculates each radius from the off-diagonal entries in a column:
$$
R_i^{\text{col}} = \sum_{j \ne i} |a_{ji}|.
$$
The column disc is
$$
D_i^{\text{col}} =
\left\{
z \in \mathbb{C} :
|z - a_{ii}| \le R_i^{\text{col}}
\right\}.
$$
Row discs and column discs can look different, but both are valid eigenvalue bounds. In practice, it is often helpful to check both modes and see which one gives tighter or more informative regions.
For real matrices, the diagonal entries are real, so the disc centers lie on the real axis. The discs may still extend into the upper and lower half of the complex plane because eigenvalues of a real matrix can be complex.
Row Discs, Column Discs, and Real-Axis Intervals
A Gershgorin disc is a two-dimensional region in the complex plane. Its real-axis interval is only the horizontal span of that disc:
$$
[a_{ii} - R_i,\ a_{ii} + R_i].
$$
This interval is useful for reading the graph and comparing bounds, but it is not the whole disc. A point with a nonzero imaginary part may still lie inside the disc if its distance from the center is no more than the radius.
For example, if a disc has center \(4\) and radius \(2\), its real-axis interval is
$$
[4 - 2,\ 4 + 2] = [2,\ 6].
$$
The full disc also includes complex points above and below the real axis, such as \(4 + i\), because
$$
|4+i-4| = |i| = 1 \le 2.
$$
Isolated Clusters and What They Mean
Gershgorin discs become especially informative when some discs are separated from the others.
If an isolated connected cluster contains \(k\) discs and does not touch or overlap the remaining discs, then that cluster contains exactly \(k\) eigenvalues, counted with algebraic multiplicity. This does not always tell you the exact eigenvalues, but it does tell you how many eigenvalues must be in that separated region.
A single disc that is completely separate from all other discs is the simplest case. It must contain exactly one eigenvalue. That eigenvalue may not be the center of the disc, but it must lie somewhere inside the disc.
When discs overlap into one large connected cluster, the theorem still guarantees that all eigenvalues lie in the union of the discs. The cluster rule just gives less separation information.
Examples of Gershgorin Discs in Practice
Example 1: Row Discs for a \(3 \times 3\) Matrix
Consider the matrix
$$
A =
\begin{bmatrix}
4 & 1 & -1 \\
2 & 5 & 1 \\
0 & -1 & 3
\end{bmatrix}.
$$
In row mode, the centers are the diagonal entries \(4\), \(5\), and \(3\).
For row 1, the radius is
$$
R_1 = |1| + |-1| = 2.
$$
So the first disc has center \(4\), radius \(2\), and real-axis interval
$$
[4 - 2,\ 4 + 2] = [2,\ 6].
$$
For row 2,
$$
R_2 = |2| + |1| = 3,
$$
so the interval is
$$
[5 - 3,\ 5 + 3] = [2,\ 8].
$$
For row 3,
$$
R_3 = |0| + |-1| = 1,
$$
so the interval is
$$
[3 - 1,\ 3 + 1] = [2,\ 4].
$$
The three row discs are therefore:
| Disc | Center | Radius | Real-axis interval |
|---|---|---|---|
| \(D_1\) | \(4\) | \(2\) | \([2,\ 6]\) |
| \(D_2\) | \(5\) | \(3\) | \([2,\ 8]\) |
| \(D_3\) | \(3\) | \(1\) | \([2,\ 4]\) |
Every eigenvalue of \(A\) lies in the union of these three discs.
Example 2: Column Discs Can Give Different Bounds
Using the same matrix,
$$
A =
\begin{bmatrix}
4 & 1 & -1 \\
2 & 5 & 1 \\
0 & -1 & 3
\end{bmatrix},
$$
column mode uses off-diagonal column sums instead.
For column 1,
$$
R_1 = |2| + |0| = 2.
$$
For column 2,
$$
R_2 = |1| + |-1| = 2.
$$
For column 3,
$$
R_3 = |-1| + |1| = 2.
$$
The column discs are:
| Disc | Center | Radius | Real-axis interval |
|---|---|---|---|
| \(D_1\) | \(4\) | \(2\) | \([2,\ 6]\) |
| \(D_2\) | \(5\) | \(2\) | \([3,\ 7]\) |
| \(D_3\) | \(3\) | \(2\) | \([1,\ 5]\) |
These are still valid Gershgorin bounds. They are not identical to the row discs because the off-diagonal column sums are different from the off-diagonal row sums.
Example 3: A Separated-Disc Case
Consider
$$
B =
\begin{bmatrix}
5 & 0.2 \\
-0.1 & 2
\end{bmatrix}.
$$
In row mode, the discs are:
$$
D_1: \text{ center } 5,\quad R_1 = |0.2| = 0.2
$$
and
$$
D_2: \text{ center } 2,\quad R_2 = |-0.1| = 0.1.
$$
Their real-axis intervals are
$$
[4.8,\ 5.2]
$$
and
$$
[1.9,\ 2.1].
$$
These intervals are far apart, and the full discs are separated as well. Because each disc is isolated, each disc contains exactly one eigenvalue.
Example 4: A Zero-Radius Edge Case
If a row has no off-diagonal contribution, its Gershgorin radius is zero. For a diagonal matrix,
$$
C =
\begin{bmatrix}
-2 & 0 \\
0 & 0.5
\end{bmatrix},
$$
both row radii are zero:
$$
R_1 = 0,\qquad R_2 = 0.
$$
The Gershgorin discs collapse to points at \(-2\) and \(0.5\). For a diagonal matrix, those points are exactly the eigenvalues.
How to Interpret the Result
The result is a bound, not usually an exact eigenvalue calculation. Each disc tells you a region where eigenvalues may lie, and the union of all discs contains every eigenvalue.
A smaller radius means the corresponding row or column is more dominated by its diagonal entry. Smaller discs usually give tighter eigenvalue localization. A larger radius means stronger off-diagonal influence and a wider possible eigenvalue region.
The maximum radius is a quick way to see how wide the loosest disc is. It is not a complete measure of the matrix, but it helps identify whether one row or column is creating a broad bound.
The disc table lists each center, radius, and real-axis interval. Use the table for exact numerical reading. Use the graph for geometric interpretation: overlap, separation, left-half-plane location, and whether approximate eigenvalue markers fall inside the guaranteed regions.
If the output reports isolated clusters, read them as eigenvalue-count information. An isolated cluster of \(k\) discs contains exactly \(k\) eigenvalues counted with multiplicity.
If all discs lie strictly to the left of the imaginary axis, then all eigenvalues also lie in the open left half-plane. For a continuous-time linear system model, that is a strong stability hint. If a disc crosses the imaginary axis, that does not prove instability; it only means the Gershgorin bound alone is not enough to prove that all eigenvalues have negative real parts.
If all discs lie inside the unit disk, then all eigenvalues lie inside the unit disk as well. Whether that matters depends on the problem you are studying.
Approximate eigenvalue markers, when shown, should be treated as comparison aids. The Gershgorin discs provide the guaranteed regions. The markers help you see how the approximate eigenvalues relate to those regions, but they are not the main theorem-based guarantee.
Common Mistakes and Misconceptions
Mistake 1: Treating the center as the eigenvalue.
The diagonal entry \(a_{ii}\) is the center of a disc, not necessarily an eigenvalue. A disc can contain no eigenvalue by itself unless it is isolated in the appropriate way.
Mistake 2: Confusing the radius with the diagonal entry.
The radius comes from off-diagonal absolute values. The diagonal entry gives the center.
Mistake 3: Using row sums while reading column mode.
Row mode uses sums across rows. Column mode uses sums down columns. The two modes can produce different discs.
Mistake 4: Forgetting the absolute values.
Negative off-diagonal entries still add positive distance to the radius. For example, \(|-3| = 3\).
Mistake 5: Rounding too early.
If you round entries or radii before comparing discs, you may accidentally make two discs appear separated or overlapping when the exact values say otherwise. The editable grid preserves accepted finite entries when it is rebuilt; rounding is used only for displayed derived values.
Mistake 6: Reading the real-axis interval as the whole disc.
The interval \([a_{ii}-R_i,\ a_{ii}+R_i]\) is only the horizontal span. The disc also includes complex points above and below that interval.
Mistake 7: Assuming overlap means every disc contains an eigenvalue.
Overlapping discs may share eigenvalue regions. The theorem guarantees that the union contains all eigenvalues, not that each individual non-isolated disc contains one.
Mistake 8: Overinterpreting approximate eigenvalue markers.
For larger non-symmetric matrices, the calculator withholds the overlay because a real-only marker would not represent complex eigenvalues reliably. When markers are shown, use them as visual aids, not as the theorem itself.
When to Use Gershgorin Discs
Use Gershgorin discs when you want a fast, interpretable eigenvalue bound for a square matrix.
They are useful for:
- checking possible eigenvalue locations before computing eigenvalues directly;
- studying matrices that are close to diagonal;
- understanding diagonal dominance;
- estimating whether eigenvalues may lie away from zero;
- finding separated eigenvalue regions;
- teaching the relationship between matrix entries and eigenvalue location;
- getting a quick stability hint for systems where eigenvalue location matters;
- comparing row-based and column-based eigenvalue bounds.
They are less useful when you need exact eigenvalues, eigenvectors, condition numbers, or highly precise spectral information.
Limitations and Things to Keep in Mind
Gershgorin’s theorem gives inclusion regions. It does not normally give exact eigenvalues.
A large disc may be a weak bound. The true eigenvalue could be much closer to the center than the radius suggests, but the theorem only guarantees the larger region.
Overlapping discs can reduce the amount of information you get. If all discs overlap into one connected region, the theorem still contains all eigenvalues, but it may not separate them well.
The row and column versions are both valid, but neither is always best. One matrix may give tighter row discs, while another may give tighter column discs.
For real matrices, complex eigenvalues can appear in conjugate pairs. A graph helps show possible complex-plane locations, but the real-axis intervals alone are not enough to describe complex eigenvalue regions.
This calculator accepts real square matrices from \(2 \times 2\) through \(6 \times 6\). It does not accept rectangular matrices, empty cells, complex entries such as \(1+2i\), nonnumeric values, \(\text{NaN}\), or infinite values.
The graph and table format displayed numbers for readability. Exact numeric zero displays as \(0\); small nonzero values and very large values may display in exponential notation. This formatting does not rewrite accepted matrix entries when the editable grid is rebuilt. Avoid making mathematical decisions from rounded display values when the exact distinction matters.
The optional eigenvalue overlay is a convenience feature. For \(2 \times 2\) matrices, it uses a scaled quadratic calculation and is withheld if a finite result cannot be resolved. For exactly symmetric matrices, it uses a scale-normalized Jacobi approximation. For general non-symmetric matrices larger than \(2 \times 2\), the overlay is withheld because a real-only marker would not represent complex eigenvalues reliably.
When eigenvalue bounds affect engineering, safety, financial, or scientific decisions, verify the result with a suitable eigenvalue method and consult a qualified professional when appropriate.
How to Use This Calculator
- Choose row discs or column discs.
- Enter a real square matrix in the grid, or use the example, import, or resize controls.
- Keep the matrix size between \(2 \times 2\) and \(6 \times 6\).
- Turn on the approximate eigenvalue overlay only if you want visual comparison markers.
- Read the result summary to see the number of discs and isolated clusters.
- Use the disc table to compare each center, radius, and real-axis interval.
- Inspect the graph to see where the discs lie in the complex plane.
- Hover on the graph when available to check coordinates, nearby centers, eigenvalue markers, and disc membership.
- Use the PNG download button if you want to save the rendered graph.
For pasted matrix data, use one row per line. Entries may be separated by spaces, commas, semicolons, or tabs. The imported matrix must be square and must contain only finite real numbers.
Frequently Asked Questions
Do Gershgorin discs give exact eigenvalues?
Usually, no. Gershgorin discs give guaranteed regions that contain the eigenvalues. They become exact in some special cases, such as diagonal matrices where every off-diagonal entry is zero.
Should I use row discs or column discs?
Both are valid. Row discs use off-diagonal row sums, while column discs use off-diagonal column sums. Try both when you want the tightest or most informative bound.
What does it mean if one disc is isolated?
An isolated disc is separate from all the other discs. In that case, it contains exactly one eigenvalue, counted with algebraic multiplicity. More generally, an isolated cluster of \(k\) discs contains exactly \(k\) eigenvalues.
Can a Gershgorin disc have radius zero?
Yes. A radius is zero when all off-diagonal entries in that selected row or column are zero. The disc then collapses to a single point at the diagonal entry.
Does a disc crossing the imaginary axis prove instability?
No. It only means the Gershgorin theorem alone cannot prove that all eigenvalues are in the left half-plane. The matrix may still have stable eigenvalues, but you need a sharper bound or a direct eigenvalue calculation to know.
Are approximate eigenvalue markers the same as Gershgorin discs?
No. Gershgorin discs are guaranteed inclusion regions from the theorem. Approximate eigenvalue markers are numerical comparison points that help you see where computed eigenvalue estimates lie relative to the discs.
Sources and References
Books
- Richard S. Varga. Geršgorin and His Circles. Springer Series in Computational Mathematics, vol. 36, Springer Berlin, Heidelberg, 2004. Basic Theory, pp. 1–33; Geršgorin-type eigenvalue inclusion chapters. Springer Nature
- Roger A. Horn and Charles R. Johnson. Matrix Analysis. 2nd ed., Cambridge University Press, 2013. Matrix analysis and eigenvalue localization background. Google Books preview
- Gene H. Golub and Charles F. Van Loan. Matrix Computations. 4th ed., Johns Hopkins University Press, 2013. Chapters 7–8 on unsymmetric and symmetric eigenvalue problems, including QR and Jacobi methods. Johns Hopkins University Press
Online and Educational Sources
- Nicholas J. Higham. “What Is Gershgorin’s Theorem?” Nick Higham, November 22, 2022. https://nhigham.com/2022/11/22/what-is-gershgorins-theorem/
- Eric W. Weisstein. “Gershgorin Circle Theorem.” MathWorld—A Wolfram Web Resource. Accessed June 28, 2026. https://mathworld.wolfram.com/GershgorinCircleTheorem.html
- Daniel Katzman, Jessica Moreno, Jason Noelanders, and Mark Winston-Galant. “10.4: Using Eigenvalues and Eigenvectors to Find Stability and Solve ODEs.” Engineering LibreTexts, last updated October 12, 2024. https://eng.libretexts.org/Bookshelves/IndustrialandSystemsEngineering/ChemicalProcessDynamicsandControls%28Woolf%29/10%3ADynamicalSystemsAnalysis/10.04%3AUsingeigenvaluesandeigenvectorstofindstabilityandsolve_ODEs