Trace
Sum of the diagonal entries; equals sum of eigenvalues.
Key idea
Sum of the diagonal entries; equals sum of eigenvalues.
In depth
The trace is the sum of the diagonal entries; it equals the sum of the eigenvalues. It is linear and satisfies tr(AB) = tr(BA), so it is unchanged by a change of basis.
Worked example
tr [4 1; 2 3] = 7, equal to eigenvalues 2 + 5.
Formula
Real-world uses
- Measuring total variance in statistics
- Checking eigenvalue sums in physics and quantum computing
Common mistake
Thinking tr(AB) = tr(A)tr(B).
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 Trace?
Sum of the diagonal entries; equals sum of eigenvalues. It is a topic at the expert level in Linear Algebra.
What is the formula for Trace?
The key formula is tr(A) = Σλᵢ.
Where is Trace used in real life?
Real-world examples: Measuring total variance in statistics; Checking eigenvalue sums in physics and quantum computing.
What should I learn before Trace?
Make sure you are comfortable with Pseudoinverse, the previous topic in Linear Algebra.
What is a common mistake with Trace?
Thinking tr(AB) = tr(A)tr(B).
How can I practise Trace?
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 Trace?
Move on to Cayley–Hamilton Theorem 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.