Trace of a Square Matrix - Python for Integrated Circuits - - An Online Book - |
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Python for Integrated Circuits http://www.globalsino.com/ICs/ | ||||||||
Chapter/Index: Introduction | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | Appendix | ||||||||
================================================================================= 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,
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:
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: The trace of A, Tr(A), is then defined as the sum of the diagonal elements of the matrix: 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: The trace of the transpose of A, which is Aᵀ, will also be the sum of its diagonal elements: 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): Let's calculate the derivative of f(A) = Tr(AB) with respect to A: 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: Now, let's compute the derivative: Using the properties of the trace and differentiating matrix products: 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, ============================================
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