trace of a matrix

Let A=(ai,j) be a square matrixMathworldPlanetmath of order n. The trace of the matrix is the sum of the main diagonal:


The trace of a matrix A is also commonly denoted as Tr(A) or TrA.


  1. 1.

    The trace is a linear transformation from the space of square matrices to the real numbers. In other words, if A and B are square matrices with real (or complex) entries, of same order and c is a scalar, then

    trace(A+B) = trace(A)+trace(B),
    trace(cA) = ctrace(A).
  2. 2.

    For the transposeMathworldPlanetmath and conjugate transposeMathworldPlanetmath, we have for any square matrix A with real (or complex) entries,

    trace(At) = trace(A),
    trace(A) = trace(A)¯.
  3. 3.

    If A and B are matrices such that AB is a square matrix, then


    For this reason it is possible to define the trace of a linear transformation, as the choice of basis does not affect the trace. Thus, if A,B,C are matrices such that ABC is a square matrix, then

  4. 4.

    If B is in invertiblePlanetmathPlanetmath square matrix of same order as A, then


    In other words, the trace of similar matricesMathworldPlanetmath are equal.

  5. 5.

    Let A be a square matrix of order n with real (or complex) entries aij. Then

    traceAA = traceAA
    = i,j=1n|aij|2.

    Here is the complex conjugateMathworldPlanetmath, and || is the complex modulusMathworldPlanetmath. In particular, traceAA0 with equality if and only if A=0. (See the Frobenius matrix norm.)

  6. 6.

    Various inequalities for trace are given in [2].

See the proof of properties of trace of a matrix.


  • 1 The Trace of a Square Matrix. Paul Ehrlich, [online] ehrlich/trace.html ehrlich/trace.html
  • 2 Z.P. Yang, X.X. Feng, A note on the trace inequality for products of Hermitian matrixMathworldPlanetmath power, Journal of Inequalities in Pure and Applied Mathematics, Volume 3, Issue 5, 2002, Article 78,
Title trace of a matrix
Canonical name TraceOfAMatrix
Date of creation 2013-03-22 11:59:56
Last modified on 2013-03-22 11:59:56
Owner Daume (40)
Last modified by Daume (40)
Numerical id 20
Author Daume (40)
Entry type Definition
Classification msc 15A99
Related topic ShursInequality