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2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: : R +R a11 a12 a13 ain ... a21 a22 a23 a2n ... a32 Ha11 + a22 + a33 +...+ ann = > ai. a31 a33 ... i=1 an2 An3 ann ... anl Let In be the n x n identity matrix (or equivalently, a vector in R"). Calculate (A) that tr(In) = = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and brl. Then compute tr(AB) and tr(BA).

2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: : R +R a11 a12 a13 ain ... a21 a22 a23 a2n ... a32 Ha11 + a22 + a33 +...+ ann = > ai. a31 a33 ... i=1 an2 An3 ann ... anl Let In be the n x n identity matrix (or equivalently, a vector in R"). Calculate (A) that tr(In) = = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and brl. Then compute tr(AB) and tr(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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2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: tr : R" +R a11 a12 a13 ain ... a22 a23 a2n ... a21 a32 a33 Ha11 + a22 + a33 + ...+ ann = ... a31 i=1 An3 ann ... anl an2 Let In be the n x n identity matrix (or equivalently, a vector in R""). Calculate (A) that tr(In) = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and bri. Then compute tr(AB) and tr(BA).
Transcribed Image Text:2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: tr : R" +R a11 a12 a13 ain ... a22 a23 a2n ... a21 a32 a33 Ha11 + a22 + a33 + ...+ ann = ... a31 i=1 An3 ann ... anl an2 Let In be the n x n identity matrix (or equivalently, a vector in R""). Calculate (A) that tr(In) = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and bri. Then compute tr(AB) and tr(BA).
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