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Definition: The trace of A E Cnxn denoted as Trace(A) is the sum of diagonal elements Trace(A) =an i=1 Note that trace function is linear. So that, for any A, B e Cnxn (i) Trace(a A) = a Trace(A) for any a E C (ii) Trace(A + B) = Trace(A) + Trace(B). Proposition: For A E Cmxn and BE Cnxm Trace(AB)=Trace(BA).

Definition: The trace of A E Cnxn denoted as Trace(A) is the sum of diagonal elements Trace(A) =an i=1 Note that trace function is linear. So that, for any A, B e Cnxn (i) Trace(a A) = a Trace(A) for any a E C (ii) Trace(A + B) = Trace(A) + Trace(B). Proposition: For A E Cmxn and BE Cnxm Trace(AB)=Trace(BA).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you please show the proof of this theorem?

Definition: The trace of A € Cnxn denoted as Trace(A) is the sum of diagonal elements: Trace(A) =ai i=1 Note that trace function is linear. So that, for any A, B e Cnxn (i) Trace(a A) = a Trace(A) for any a E C (ii) Trace(A + B) = Trace(A) + Trace(B). Proposition: For A E Cmxn and B E CnXm Trace(AB)=Trace(BA).
Transcribed Image Text:Definition: The trace of A € Cnxn denoted as Trace(A) is the sum of diagonal elements: Trace(A) =ai i=1 Note that trace function is linear. So that, for any A, B e Cnxn (i) Trace(a A) = a Trace(A) for any a E C (ii) Trace(A + B) = Trace(A) + Trace(B). Proposition: For A E Cmxn and B E CnXm Trace(AB)=Trace(BA).
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