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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.

1811 questions
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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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73
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7 answers

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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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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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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How to prove $\det \left(e^A\right) = e^{\operatorname{tr}(A)}$?

Prove $$\det \left( e^A \right) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}^{n \times n}$.
John
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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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Coordinate free proof that $\operatorname{trace}(A)=0\:\Longrightarrow\:A=BC-CB$

As you probably know, the trace function on square matrices has the property that $$\operatorname{trace}(AB-BA)=0\,.$$ You might also know that the converse is true: $$\operatorname{trace}(A)=0\;\text{ implies } A=BC-CB\:\text{ for some matrices }…
Gregory Grant
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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 of an inverse matrix

I want to know if there is a way to simplify or a closed form solution of $\operatorname{tr} \left( \Sigma^{-1} \right)$, where $\Sigma$ is a symmetric positive definite matrix.
sachinruk
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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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2 answers

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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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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$\operatorname{tr}(AABABB) = \operatorname{tr}(AABBAB)$ 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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Does the set of matrix commutators form a subspace?

The following is an interesting problem from Linear Algebra 2nd Ed - Hoffman & Kunze (3.5 Q17). Let $W$ be the subspace spanned by the commutators of $M_{n\times n}\left(F\right)$: $$C=\left[A, B\right] = AB-BA$$ Prove that $W$ is exactly the…
EuYu
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