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Questions tagged [trace]

For questions about trace, which can concern matrices, operators or functions. If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.

In the context of linear algebra, the trace of a square matrix $M$ is the sum of the diagonal entries.

It's almost the same idea in the case of operators on a separable Hilbert space (with conditions of convergence).

In the context of partial differential equations, when we work with an open set having good conditions, we can define what we call a trace operator.

Use it with the appropriate tags.

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Proof that the trace of a matrix is the sum of its eigenvalues

I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
JohnK
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The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow
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Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?

I am searching for a short coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ for linear operators $A$, $B$ between finite dimensional vector spaces of the same dimension. The usual proof is to represent the operators as…
Potato
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Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \det(A)\det(B) && \mathrm{tr}(A+B)= \mathrm{tr}(A)…
Mike F
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Proving: "The trace of an idempotent matrix equals the rank of the matrix"

How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"? This is another property that is used in my module without any proof, could anybody tell me how to prove this one?
Quixotic
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How to prove $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$?

there is a similar thread here Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?, but I'm only looking for a simple linear algebra proof.
athos
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$\operatorname{tr}(AABABB) = \operatorname{tr}(AABBAB)$ for $2×2$ matrices

Similar to a previous question here, I wonder if cyclic permutations are the only relations amongst traces of (non-commutative) monomials. Since the evaluations $\operatorname{tr}:k\langle x,y,\dots \rangle \to k$ take an infinite dimensional…
Jack Schmidt
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How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
John
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Direct proof that nilpotent matrix has zero trace

Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
Norbert
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How do you prove that $tr(B^{T} A )$ is a inner product?

Consider the vectorspace of all real $m \times n$ vectors and define an inner product $\langle A,B\rangle = \operatorname{tr}(B^T A)$. "tr" stands for "trace" which is the sum of the diagonal entries of a matrix. How do you prove…
Dieter Verbeemen
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'Trace trick' for expectations of quadratic forms

I am trying to understand the proof for the Kullback-Leibler divergence between two multivariate normal distributions. On the way, a sort of trace trick is applied for the expectation of the quadratic form $$E[ (x-\mu)^T \Sigma^{-1} (x-\mu) ]=…
tomka
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Trace of an Inverse Matrix

I want to know if there is a way to simplify, or a closed form solution of $tr(\Sigma^{-1})$ where $\Sigma$ is a symmetric positive definite matrix.
sachinruk
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Is trace invariant under cyclic permutation with rectangular matrices?

I'm working with trace of matrices. Trace is defined for square matrix and there are some useful rule, i.e. $\text{tr}(AB) = \text{tr}(BA)$, with $A$ and $B$ square, and more in general trace is invariant under cyclic permutation. I was wondering if…
the_candyman
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Symmetric Matrices with trace zero

Let $M_n$ denote the set of complex matrices of order $n$. It is well known that if $A\in M_n$ has trace zero then $A$ can be written as $A=BC-CB$, where $B,C\in M_n$. Is it true that every symmetric matrix $S\in M_n$ with trace zero can be written…
Daniel
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Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the Wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite dimensional vector space over a field $F$ and define a…
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