Gershgorin Circle Theorem
Bounds on eigenvalue locations.
Key idea
Bounds on eigenvalue locations.
In depth
Every eigenvalue of A lies in a disc centred at some diagonal entry aᵢᵢ with radius equal to the sum of absolute off-diagonal entries in that row. It gives quick eigenvalue bounds.
Worked example
For [4 1; 1 3], discs at 4 (radius 1) and 3 (radius 1), so eigenvalues lie in [2, 5].
Real-world uses
- Estimating where eigenvalues lie for stability checks without full calculation
Common mistake
Using columns and rows inconsistently when finding the radius.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Gershgorin Circle Theorem?
Bounds on eigenvalue locations. It is a topic at the expert level in Linear Algebra.
Where is Gershgorin Circle Theorem used in real life?
Real-world examples: Estimating where eigenvalues lie for stability checks without full calculation.
What should I learn before Gershgorin Circle Theorem?
Make sure you are comfortable with Power Iteration, the previous topic in Linear Algebra.
What is a common mistake with Gershgorin Circle Theorem?
Using columns and rows inconsistently when finding the radius.
How can I practise Gershgorin Circle Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Gershgorin Circle Theorem?
Move on to Linear Maps Between Spaces next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.