In linear algebra, the trace of a square matrix A, denoted tr (A), [1] is the sum of the elements on its main diagonal, . It is only defined for a square matrix (n × n). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, tr …

2022年9月17日 · Given a matrix A A, we can “find the trace of A A,” which is not a matrix but rather a number. We formally define it here. Let A A be an n × n n × n matrix. The trace of A A, denoted tr(A) tr (A), is the sum of the diagonal elements of A A. That is, tr(A) = a11 +a22 + ⋯ +ann. tr (A) = a 11 + a 22 + ⋯ + a n n.

2025年1月2日 · The trace of a matrix, denoted as tr(A), is the sum of its diagonal elements and has key properties such as linearity, invariance under transposition, and relationships with scalar multiplication and matrix products.

In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then. where tr (X) is the trace of the matrix X and is its adjugate matrix.

矩阵的迹，定义为方阵对角线上元素之和，即 tr (A)=a_ {11}+...+a_ {nn}=\sum\limits_ {i=1}^ {n} a_ {ii}. 矩阵的迹等于矩阵的特征值之和。 证明：若方阵A可以进行特征分解，方阵 Q 的各列为A的特征向量， \Lambda 为元素为对应特征值组成的对角矩阵， \therefore tr (A)=tr (Q\Lambda Q^ {-1})=tr (\Lambda Q^ {-1}Q)=tr (\Lambda) 若A是实对称矩阵，则 A=tr (Q\Lambda Q^ {T})

4 天之前 · The trace of an n×n square matrix A is defined to be Tr (A)=sum_ (i=1)^na_ (ii), (1) i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr [list]. In group theory, traces are known as "group characters."

For $2\times 2$ matrices, the answer is $(\operatorname{tr} A)^2-2\det A$. For an $n\times n$ matrix, the characteristic polynomial includes the trace, determinant and $n-2$ other functions in between.

2022年11月29日 · Definition: The symbol that will be used for the trace function in this paper is Tr (). Thus, for the n × n matrix A, Tr (A) ≡ a11 + a22 + · · · + ann , that is, the trace is the sum of the components on the main diagonal. Figure 1. Let A be an n × n matrix.

Let A ⊗ B A ⊗ B denote the tensor product of two matrices, A A and B B. I can show the trace of it is the same as the product of the traces of A A and B B, which follows from computation. Is there some conceptual explanation for this?

Here is the theorem about traces. Theorem. The following properties of traces hold: Proof. Properties 1,2 and 3 immediately follow from the definition of the trace. Let us prove the fourth property: The trace of AB is the sum of diagonal entries of this matrix. By the definition of the product of two matrices, these entries are: