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In mathematics, the trace of a square matrix is a scalar value that represents the sum of the diagonal elements of the matrix. The trace is often denoted as "Tr" followed by the matrix's name or symbol. If A is an n x n square matrix, the trace of A is given by:

Tr(A) = a₁₁ + a₂₂ + a₃₃ + ... + aₙₙ ------------------------ [3896a]

where,

• a₁₁, a₂₂, a₃₃, ..., aₙₙ are the diagonal elements of the matrix A, which are the elements with the same row and column index.

The trace of a matrix is a simple but important concept in linear algebra and is used in various mathematical and computational contexts. Some key properties and uses of the trace include:

1. Linearity: The trace is a linear operator, which means that for matrices A and B of the same size and scalars α and β, Tr(αA + βB) = αTr(A) + βTr(B).

2. Invariance: The trace is invariant under similarity transformations. That is, if A and B are similar matrices (i.e., B = P⁻¹AP for some invertible matrix P), then Tr(A) = Tr(B).

3. Characteristic Polynomial: The trace is related to the coefficients of the characteristic polynomial of a matrix, which helps in finding eigenvalues.

4. Matrix Norm: The Frobenius norm of a matrix, which is a way to measure the "size" of a matrix, is related to the trace. Specifically, ||A||F = sqrt(Tr(AAᵀ)).

5. Quadratic Forms: The trace is used in expressing quadratic forms involving matrices.

We can know, for a square matrix A (an n x n matrix), the trace of A, denoted as Tr(A), will be equal to the sum of its diagonal entries.

Assuming the square matrix A is:

          ------------------------ [3896b]

The trace of A, Tr(A), is then defined as the sum of the diagonal elements of the matrix:

          ------------------------ [3896c]

The trace of a matrix A is equal to the trace of its transpose, denoted as Tr(A) = Tr(Aᵀ). This is a property of matrix traces and can be easily shown using the definition of the trace.

The trace of a matrix A is defined as the sum of its diagonal elements:

          ------------------------ [3896d]

The trace of the transpose of A, which is Aᵀ, will also be the sum of its diagonal elements:

          ------------------------ [3896e]

However, the transpose of a matrix doesn't change the values on its main diagonal; it only changes the arrangement of elements in rows and columns. Therefore, (a₁₁)ᵀ is still equal to a₁₁, (a₂₂)ᵀ is still equal to a₂₂, and so on. Consequently, Tr(Aᵀ) is equal to Tr(A):

          ------------------------ [3896f]

Let's calculate the derivative of f(A) = Tr(AB) with respect to A:

          ------------------------ [3896g]

Now, we want to find ∂f/∂A, the derivative of f with respect to A.

Using the trace properties, we can rewrite Tr(AB) as the trace of the product BA because the trace of a product of matrices is invariant under cyclic permutation:

          ------------------------ [3896h]

Now, let's compute the derivative:

          ------------------------ [3896i]

Using the properties of the trace and differentiating matrix products:

          ------------------------ [3896j]

Therefore, the derivative of f(A) with respect to A is indeed B^T (the transpose of matrix B). This is a fundamental result in matrix calculus and is often used in various mathematical and computational contexts, including optimization and machine learning.

And more equations related to trace are,

          ------------------------ [3896k]

          ------------------------ [3896l]

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