proof of properties of trace of a matrix

Proof of Properties:

  1. 1.

    Let us check linearity. For sums we have

    trace(A+B) = i=1n(ai,i+bi,i)   (property of matrix addition)
    = i=1nai,i+i=1nbi,i(property of sums)
    = trace(A)+trace(B).


    trace(cA) = i=1ncai,i(property of matrix scalar multiplication)
    = ci=1nai,i(property of sums)
    = ctrace(A).
  2. 2.

    The second property follows since the transposeMathworldPlanetmath does not alter the entries on the main diagonal.

  3. 3.

    The proof of the third property follows by exchanging the summation order. Suppose A is a n×m matrix and B is a m×n matrix. Then

    traceAB = i=1nj=1mAi,jBj,i
    = j=1mi=1nBj,iAi,j(changing summation order)
    = traceBA.
  4. 4.

    The last property is a consequence of Property 3 and the fact that matrix multiplicationMathworldPlanetmath is associative;

    trace(B-1AB) = trace((B-1A)B)
    = trace(B(B-1A))
    = trace((BB-1)A)
    = trace(A).
Title proof of properties of trace of a matrix
Canonical name ProofOfPropertiesOfTraceOfAMatrix
Date of creation 2013-03-22 13:42:54
Last modified on 2013-03-22 13:42:54
Owner Daume (40)
Last modified by Daume (40)
Numerical id 4
Author Daume (40)
Entry type Proof
Classification msc 15A99