10. (In this problem we show any square matrix is the sum of a symmetric matrix and a skew-symmetric matrix). Let A be a square matrix. From this define and =(4 (A + A¹) B = C = (A - A¹). (a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e. B = B. Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e. Ct=-C. (c) Show B+C = A. (d) An example: for A = 1 2 34 , compute B and C. 11. For two nx n matrices A and B, we define the commutator, denoted [A, B], by [A, B]: = AB - BA. a) What is [A, I], where I is the n x n identity matrix? b) What is [A, A] for any n x n matrix A ? c) Give an example of two nonzero 2 x 2 matrices A and B such that [A, B] is not the zero matrix. d) Fix an n x n matrix C. Show that [A + B, C] = [A, C] + [B, C] e) [Optional but fun!] Compute out this sum [A, [B, C]] + [B, [C, A]] + [C, [A, B]].
10. (In this problem we show any square matrix is the sum of a symmetric matrix and a skew-symmetric matrix). Let A be a square matrix. From this define and =(4 (A + A¹) B = C = (A - A¹). (a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e. B = B. Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e. Ct=-C. (c) Show B+C = A. (d) An example: for A = 1 2 34 , compute B and C. 11. For two nx n matrices A and B, we define the commutator, denoted [A, B], by [A, B]: = AB - BA. a) What is [A, I], where I is the n x n identity matrix? b) What is [A, A] for any n x n matrix A ? c) Give an example of two nonzero 2 x 2 matrices A and B such that [A, B] is not the zero matrix. d) Fix an n x n matrix C. Show that [A + B, C] = [A, C] + [B, C] e) [Optional but fun!] Compute out this sum [A, [B, C]] + [B, [C, A]] + [C, [A, B]].
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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Need help solve question 11
![10. (In this problem we show any square matrix is the sum of a symmetric matrix and a
skew-symmetric matrix).
Let A be a square matrix. From this define
and
=
(d) An example: for A
=
1 2
3 4
-
B
с
-
(a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e.
Bt B.
7
2
=
1
(b) Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e.
Ct=-C.
(c) Show B+C = A.
(4+
(A + A¹)
2
1
(4-
(A – Aª).
11. For two nx n matrices A and B, we define the commutator, denoted [A, B], by
[A, B]
AB - BA.
a) What is [A, I], where I is the n x n identity matrix?
b) What is [A, A] for any n x n matrix A ?
c) Give an example of two nonzero 2 × 2 matrices A and B such that [A, B] is not the
zero matrix.
d) Fix an n x n matrix C. Show that [A + B, C] = [A, C] + [B, C]
e) [Optional but fun!] Compute out this sum
compute B and C.
[A, [B, C]] + [B, [C, A]] + [C, [A, B]].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8783cbc5-0f10-43ba-8bd7-4625ee264f91%2Fcae4772f-b41e-4edb-aebc-ef67ff9dc557%2Fxbh1ef_processed.jpeg&w=3840&q=75)
Transcribed Image Text:10. (In this problem we show any square matrix is the sum of a symmetric matrix and a
skew-symmetric matrix).
Let A be a square matrix. From this define
and
=
(d) An example: for A
=
1 2
3 4
-
B
с
-
(a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e.
Bt B.
7
2
=
1
(b) Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e.
Ct=-C.
(c) Show B+C = A.
(4+
(A + A¹)
2
1
(4-
(A – Aª).
11. For two nx n matrices A and B, we define the commutator, denoted [A, B], by
[A, B]
AB - BA.
a) What is [A, I], where I is the n x n identity matrix?
b) What is [A, A] for any n x n matrix A ?
c) Give an example of two nonzero 2 × 2 matrices A and B such that [A, B] is not the
zero matrix.
d) Fix an n x n matrix C. Show that [A + B, C] = [A, C] + [B, C]
e) [Optional but fun!] Compute out this sum
compute B and C.
[A, [B, C]] + [B, [C, A]] + [C, [A, B]].
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