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[parent] proof of properties of trace of a matrix (Proof)

Proof of Properties:

  1. Let us check linearity. For sums we have
    $\displaystyle \operatorname{trace}(A+B)$ $\displaystyle =$ $\displaystyle \sum\limits_{i=1}^{n} (a_{i,i} + b_{i,i})\,\,\,\,\,\,\,\,\,\,\,$   (property of matrix addition)  
      $\displaystyle =$ $\displaystyle \sum\limits _{i=1} ^{n} a_{i,i} + \sum\limits _{i=1} ^{n} b_{i,i}\,\,\,\,$   (property of sums)  
      $\displaystyle =$ $\displaystyle \operatorname{trace}(A) + \operatorname{trace}(B).$  

    Similarly,


    $\displaystyle \operatorname{trace}(cA)$ $\displaystyle =$ $\displaystyle \sum\limits _{i=1} ^{n} c\cdot a_{i,i}\,\,\,\,\,$   (property of matrix scalar multiplication)  
      $\displaystyle =$ $\displaystyle c\cdot \sum\limits _{i=1} ^{n} a_{i,i}\,\,\,\,\,$   (property of sums)  
      $\displaystyle =$ $\displaystyle c\cdot \operatorname{trace}(A).$  

  2. The second property follows since the transpose does not alter the entries on the main diagonal.
  3. The proof of the third property follows by exchanging the summation order. Suppose $ A$ is a $ n\times m$ matrix and $ B$ is a $ m\times n$ matrix. Then
    $\displaystyle \operatorname{trace} AB$ $\displaystyle =$ $\displaystyle \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{m} A_{i,j} B_{j,i}$  
      $\displaystyle =$ $\displaystyle \sum\limits_{j=1}^{m} \sum\limits_{i=1}^{n} B_{j,i} A_{i,j} \,\,\,\,$(changing summation order)  
      $\displaystyle =$ $\displaystyle \operatorname{trace} BA.$  

  4. The last property is a consequence of Property 3 and the fact that matrix multiplication is associative;
    $\displaystyle \operatorname{trace} (B^{-1} A B)$ $\displaystyle =$ $\displaystyle \operatorname{trace} \big((B^{-1} A) B\big)$  
      $\displaystyle =$ $\displaystyle \operatorname{trace} \big(B(B^{-1} A) \big)$  
      $\displaystyle =$ $\displaystyle \operatorname{trace} \big( (BB^{-1}) A \big)$  
      $\displaystyle =$ $\displaystyle \operatorname{trace} (A).$  



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Cross-references: associative, matrix multiplication, consequence, matrix, order, diagonal, transpose, sums, properties, proof
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This is version 1 of proof of properties of trace of a matrix, born on 2003-06-25.
Object id is 4396, canonical name is ProofOfPropertiesOfTraceOfAMatrix.
Accessed 20803 times total.

Classification:
AMS MSC15A99 (Linear and multilinear algebra; matrix theory :: Miscellaneous topics)

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