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.

Matrix Trace Visualizer | Sum of Diagonal Entries

Key Terms

Trace — the sum of the main-diagonal entries of a square matrix:

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

.

Main diagonal — the entries

a_{i,i}

where the row index equals the column index.

Square matrix — a matrix with the same number of rows and columns (

n \times n

). Trace is defined only for square matrices.

Off-diagonal entries — entries

a_{i,j}

with

i \neq j

. They are completely ignored by the trace.

Scalar invariant — the trace returns a single number that is invariant under similarity transformations:

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

.

$\Sigma$ notation — the trace can be written compactly as

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

.

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

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

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

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.

Scene 0: the Question, Before Anything Is Highlighted

The player opens on a deliberately blank slate. The matrix

A

is drawn with every cell in its neutral grey, no diagonal marked, no running sum on screen — just the question of what the trace of this matrix is.

That restraint is the point. Before any procedure runs, the only thing established is that

A

is square,

4 \times 4

at the default dimension, and that a single number is about to be extracted from it.

Scene 0, frozen

Sixteen neutral cells and nothing else - no diagonal marked, no running sum. The only fact established so far is that A is square, which is the one thing the trace requires.

Squareness is the one precondition worth dwelling on. The trace is defined only for square matrices, because it needs entries where the row index and the column index agree — and in a

3 \times 5

matrix there is no

a_{4,4}

to reach for.

That is also why the dimension control offers a single number rather than a pair. Changing it from 4 to 7 rebuilds the scene list with three more sweep steps, but it can never produce a non-square matrix to take the trace of.

Scene 1: Revealing the Main Diagonal

The next scene turns exactly four cells blue —

a_{1,1}

,

a_{2,2}

,

a_{3,3}

and

a_{4,4}

— and leaves the other twelve grey.

Those four are the main diagonal: the cells whose row index equals their column index. Everything the trace does happens on them, and the twelve grey cells play no part in the calculation at all.

Scene 1, frozen

The four cells where the row index equals the column index turn blue. The other twelve stay grey and take no part in the calculation.

A

4 \times 4

matrix holds sixteen numbers and the trace consults four of them, so it throws away three quarters of the matrix. That is a severe compression, and it is fair to ask what survives it.

Quite a lot, as it turns out. The trace is unchanged by transposing, since transposing swaps

a_{i,j}

with

a_{j,i}

and leaves

a_{i,i}

exactly where it was. It also equals the sum of the eigenvalues — so this fixed handful of cells encodes something about the matrix as a transformation, not merely about its bookkeeping. The key properties section takes that further.

The Sweep: One Diagonal Entry at a Time

The middle scenes walk the diagonal from top-left to bottom-right, one cell per step, and the colouring carries three distinct meanings at once:

solid green — already added to the running sum
solid blue, slightly enlarged — the entry being added right now
dashed green outline — on the diagonal, still to come

The frozen picture below is the second sweep step. One entry is behind it, one is current, and two are still pending, so all three states are visible together.

Sweep step 2 of 4, frozen

Three states at once: a11 solid green (already counted), a22 blue and slightly enlarged (being added now), a33 and a44 dashed (still pending).

The formula line beneath the matrix grows in step with the colouring, so the sum is assembled in front of you rather than presented finished. At this point it reads

a_{1,1} + a_{2,2}

, with the remaining terms greyed until their turn.

Nothing here depends on the order. Addition is commutative, so sweeping bottom-right to top-left, or in any order at all, produces the same total — the left-to-right walk is a presentational choice, not part of the definition.

The Completed Trace

The final scene turns all four diagonal cells solid green and states the result in closed form:

\operatorname{tr}(A) = \sum_i a_{i,i}

with every off-diagonal entry of

A

formally ignored.

Final scene, frozen

All four diagonal cells solid green. The caption states the closed form, sum over i of a(i,i), with every off-diagonal entry formally ignored.

Written as a sum over

i

, the definition is size-independent: it reads identically whether the matrix is

2 \times 2

or

10 \times 10

, which is exactly what the dimension control demonstrates when you change

n

and replay.

The compact form is also what makes the algebraic properties easy to check. Linearity,

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

, follows immediately because addition is entrywise and the diagonal of a sum is the sum of the diagonals. The cyclic property

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

takes a little more work but comes from the same summation, and it is the reason the trace is invariant under a change of basis.

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.

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.

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.

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.

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

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.

Trace of a Matrix

Step explanations