Matrix Trace Calculator

Omni's matrix trace calculator is here to help you learn about the important mathematical concept of the trace of a matrix. Although it's not hard to learn how to calculate the trace of a matrix, it is important to understand the theoretical properties of the matrix trace. After reading the article below, you'll be able to answer the following questions:

• What is the trace of a matrix?
• What properties does the matrix trace have?
• What's the connection between trace and eigenvalues?
• Is the trace of a matrix a linear transformation?
• ... and many more!

Let's go!

We define the trace of a matrix as the sum of all the diagonal elements of this matrix. The matrix in question must be square, i.e., it must have as many columns as it has rows. For instance, the trace of the $2 × 2$ matrix, $A$,

is equal to $1 + 4 = 5$. The trace of matrix $A$ is denoted by $\operatorname{tr}(A)$.

As you can see, it's not difficult to learn how to calculate the trace of a matrix. Let's now talk about the properties of the trace.

If $A$, $B$, and $C$ are all square matrices, then their traces satisfy the following properties:

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

• $\operatorname{tr}(k A) = k \operatorname{tr}(A)$, where $k$ is a scalar. You can visit the matrix by scalar calculator if you're unsure about this type of multiplication.

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

• $\operatorname{tr}(ABC) = \operatorname{tr}(BCA) = \operatorname{tr}(CAB)$

• $\operatorname{tr}(A^T) = \operatorname{tr}(A)$, where $A^T$ is the matrix transpose of $A$

• $\operatorname{tr}(A \otimes B) = \operatorname{tr}(A) \operatorname{tr}(B)$, where $\otimes$ denotes the tensor product (aka the Kronecker product) of matrices.

• $\operatorname{tr}(A)$ is equal to the sum of the eigenvalues of $A$.

• $\operatorname{tr}(I_n) = n$, where $I_n$ is the $n \times n$ identity matrix.

The trace is famous for its cyclic property. It basically means that the trace is invariant under cyclic permutations — the cyclical rearrangement of the matrices in a matrix product expression. For instance, for three matrices $A$, $B$, and $C$, we have

and for four matrices we have

Do you see the pattern? In general, for $n$ matrices, we get

This is the cyclic property of the trace operator. It's good to remember this properly as it is often very useful in various math problems with matrix trace.

To calculate the trace of a matrix by hand, you need to:

1. Write down the coefficients of the matrix.
2. Identify the diagonal entries — the diagonal going from the upper-left corner to the bottom-right corner.
3. Add all the diagonal entries together.
4. The result you've got in Step 3 is exactly the trace of your matrix!

To verify if your computations are correct, use Omni's matrix trace calculator!

This trace of a matrix calculator is very user-friendly! To use it, you need to:

1. Choose the matrix size.
2. Enter the matrix coefficients.
3. The trace of your matrix gets calculated and displayed immediately.
4. Remember the blank fields are interpreted as zeros!

The trace of a projection matrix is equal to the dimension of the space that the matrix projects onto (the target space). This is a special case of a more general result that the trace of an idempotent matrix (recall A is called idempotent when A² = A) is equal to the rank of this matrix.

Yes, the trace operation is linear. That is, it satisfies tr(x·A + y·B) = x × tr(A) + y × tr(B), where A and B are square matrices of the same size, and x and y are scalars.

Recall that the eigenvalues are the roots of the characteristic polynomial of a matrix. Since the characteristic polynomial of a 2×2 matrix A reads p(λ) = x² − tr(A)·λ + det(A), the formula for its eigenvalues is ½ tr(A) ± ½√(tr(A)² − 4·det(A)).