What is the trace of a matrix?

The trace of an n by n matrix A, written tr of A, is the sum of its main-diagonal entries: a sub 1,1 plus a sub 2,2 and so on up to a sub n,n. It is defined only for square matrices and ignores all off-diagonal entries. The result is a single scalar.

Can you take the trace of a non-square matrix?

No. The trace is defined only for square matrices because it sums entries along the main diagonal, and only square matrices have a full main diagonal. For a rectangular matrix, the entries where i equals j extend only through the smaller of the two dimensions, and the trace is not standardly defined.

What is the relationship between trace and eigenvalues?

For any square matrix A, the trace equals the sum of the eigenvalues counted with multiplicity. This relationship holds whether or not A is diagonalizable. Combined with the fact that the determinant equals the product of eigenvalues, the trace gives a quick scalar summary of the spectrum without diagonalizing.

Is the trace of a product commutative?

The trace satisfies the cyclic property: tr of A times B equals tr of B times A. More generally, the trace is invariant under cyclic rotations of a product, so tr of A B C equals tr of B C A equals tr of C A B. However, trace is not invariant under arbitrary permutations: tr of A B C does not generally equal tr of A C B.

Why is the trace useful?

The trace is one of the simplest scalar invariants of a matrix, easy to compute and rich in meaning. It equals the sum of eigenvalues, appears in the Frobenius norm and inner product, is invariant under similarity transformations, and shows up throughout physics, statistics, and machine learning whenever a total or sum of intrinsic quantities of a square matrix is needed.

Trace of a Matrix

Matrix Trace

Symbolic visualization of tr(A) = sum of the main-diagonal entries.

Dimension of A (square)?Trace is defined only for square matrices. The trace of an n×n matrix A is the sum of its main-diagonal entries: tr(A) = a₁,₁ + a₂,₂ + … + aₙ,ₙ. Off-diagonal entries are ignored entirely.

A4×4

a1,1

a1,2

a1,3

a1,4

a2,1

a2,2

a2,3

a2,4

a3,1

a3,2

a3,3

a3,4

a4,1

a4,2

a4,3

a4,4

Step 1 / 7

Step explanations

1Trace of a square matrix (4×4)

What is the trace of A? Trace is a single number assigned to every square matrix — we'll build it step by step.

Getting Started with the Visualizer

Reading the Scene Player

Choosing the Dimension

Getting Started with the Visualizer

Set the size of

A

and watch the trace build one diagonal entry at a time.

• Use the Dimension steppers to set the size of

A

from

2 \times 2

up to

10 \times 10

— both dimensions move together because

A

must be square
• Hover the ? icon for a reminder that trace requires a square matrix
• The scene player starts by posing the question with no highlights, then reveals the main diagonal, then sweeps the diagonal entry by entry
• Use the speed selector and step log to control the pace and review prior steps

Learn More

Reading the Scene Player

Each scene focuses on the diagonal of

A

with three visual states.

• Pending entries (not yet counted) appear with a dashed green outline
• The current entry being added is highlighted in solid blue with a slight scale-up
• Counted entries turn solid green
• Off-diagonal cells stay neutral throughout — the trace ignores them completely
• The running formula

\text{tr}(A) = a_{1,1} + a_{2,2} + \cdots

updates above with the same color coding

Learn More

Choosing the Dimension

The dimension stepper controls the size of the square matrix

A

.

• Smaller sizes (

2 \times 2

3 \times 3

) make each scene easy to follow and show how short the trace sum is
• Larger sizes (up to

10 \times 10

) demonstrate how the same rule scales — exactly

n

terms regardless of how many off-diagonal entries exist
• Cell size shrinks automatically as

n

grows so the matrix stays readable
• Both row and column steppers are linked since trace only applies to square matrices

Learn More

Scene Order

The animation follows a deliberate three-stage order.

• Pose — the matrix appears with no highlights; the question "what is the trace?" is asked first
• Reveal — the entire main diagonal is highlighted in blue, separating the entries that contribute from those that do not
• Sweep — one scene per diagonal entry, adding

a_{k,k}

to the running sum
• Outro — every diagonal entry is green and the complete formula is shown along with the

\Sigma

notation

This order separates "what is the trace looking at?" from "what does the trace compute?" — two questions that are easy to conflate.

Learn More

What the Trace Is

The trace of an

n \times n

matrix

A

is the sum of its main-diagonal entries:

\text{tr}(A) = a_{1,1} + a_{2,2} + \cdots + a_{n,n} = \sum_{i=1}^{n} a_{i,i}

Trace is defined only for square matrices. Off-diagonal entries play no role at all — the trace ignores them completely. The result is a single scalar that summarizes one piece of information about

A

, complementary to the determinant.

For comprehensive theory, see matrix operations.

Learn More

Key Properties

The trace has a short list of clean algebraic properties.

• Linearity:

\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)

and

\text{tr}(kA) = k \cdot \text{tr}(A)

• Transpose invariance:

\text{tr}(A^T) = \text{tr}(A)

— the diagonal stays put under transposition
• Cyclic property:

\text{tr}(AB) = \text{tr}(BA)

, and more generally

\text{tr}(ABC) = \text{tr}(BCA) = \text{tr}(CAB)

• Similarity invariance:

\text{tr}(P^{-1} A P) = \text{tr}(A)

— trace doesn't change under change of basis
• Sum of eigenvalues: for any square

A

\text{tr}(A) = \sum_i \lambda_i

where

\lambda_i

are the eigenvalues counted with multiplicity

The cyclic property is the workhorse — it's behind nearly every nontrivial trace identity.

Learn More

Why It Matters

The trace appears throughout mathematics and applications because it captures the sum of eigenvalues in an arithmetic form that's easy to compute.

• Linear algebra:

\text{tr}(A) = \sum \lambda_i

— read off the eigenvalue sum without diagonalizing
• Differential geometry and physics: the trace of a stress or strain tensor measures volume change; the trace of a Hamiltonian relates to partition functions
• Machine learning: trace appears in covariance summaries, Frobenius norms (

\|A\|_F^2 = \text{tr}(A^T A)

), and many regularization terms
• Inner product: the Frobenius inner product is

\langle A, B \rangle_F = \text{tr}(A^T B)

• Statistics: trace of a projection matrix counts the degrees of freedom of the projection

Anywhere a "total" or "sum of intrinsic quantities" of a square matrix is needed, the trace is the right tool.

Learn More

Worked Example

Take

A

as a

3 \times 3

matrix:

A = \begin{pmatrix} 2 & 7 & -1 \\ 0 & 5 & 4 \\ 3 & 1 & -6 \end{pmatrix}

The trace pulls out only the diagonal entries:

\text{tr}(A) = 2 + 5 + (-6) = 1

The other six entries (7,

-1

, 0, 4, 3, 1) are ignored entirely. Notice that for the same

A

, the determinant uses every entry while the trace uses only three — they capture different aspects of the matrix.

Set the visualizer to

3 \times 3

and step through to see this picking-out process animated.

Learn More

Common Mistakes

A few mistakes recur with trace.

• Trying to compute the trace of a non-square matrix — the trace is undefined for rectangular

A

because there is no full main diagonal
• Confusing trace with determinant — both are scalar summaries of a square matrix, but trace sums diagonal entries while determinant computes a signed product across all permutations
• Forgetting the cyclic property is cyclic, not commutative —

\text{tr}(AB) = \text{tr}(BA)

holds, but

\text{tr}(ABC) \neq \text{tr}(ACB)

in general
• Assuming trace equals the determinant of the diagonal — the trace is a sum, not a product
• Mixing up "diagonal" with "anti-diagonal" — trace uses entries where

i = j

, not where

i + j = n + 1

Learn More

Related Concepts

Determinant — another scalar invariant of a square matrix, equal to the product of eigenvalues.

Eigenvalues — the trace equals their sum, the determinant equals their product.

Frobenius norm — defined as

\|A\|_F = \sqrt{\text{tr}(A^T A)}

.

Frobenius inner product —

\langle A, B \rangle_F = \text{tr}(A^T B)

.

Transpose — leaves the trace unchanged:

\text{tr}(A^T) = \text{tr}(A)

.

Similarity transformation — leaves the trace invariant.

Identity matrix —

\text{tr}(I_n) = n

, since every diagonal entry equals 1.

Matrix multiplication — the cyclic property of trace (

\text{tr}(AB) = \text{tr}(BA)

) is one of the most used trace identities.

Learn More