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
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdec92e-028d-4f37-af60-841922a4cee5%2Fdd94c591-b732-4b99-8070-32c7c110adab%2F03b1di_processed.png&w=3840&q=75)
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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