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With each m xn matrix Y we associate the column vector vec(Y) of length m x n defined by vec(Y) := (y11,..., Yml, Y12,-.-, Ym2, ..., Yın,..., Ymn)". (i) Let A be an m x n matrix, B an p x q matrix, and C an m x q matrix. Let X be an unknown n x p matrix. Show that the matrix equation AXB = C is equivalent to the system of qm equations in np unknowns given by (B" ® A)vec(X) = vec(C) that is, vec(AXB) = (BT ® A)vec(X). (ii) Let A, B, D be n x n matrices and In the n x n identity matrix. Use the result from (i) to prove that AX +XB = D can be written as (In ® A) + (B" ® In))vec(X) = vec(D).

With each m xn matrix Y we associate the column vector vec(Y) of length m x n defined by vec(Y) := (y11,..., Yml, Y12,-.-, Ym2, ..., Yın,..., Ymn)". (i) Let A be an m x n matrix, B an p x q matrix, and C an m x q matrix. Let X be an unknown n x p matrix. Show that the matrix equation AXB = C is equivalent to the system of qm equations in np unknowns given by (B" ® A)vec(X) = vec(C) that is, vec(AXB) = (BT ® A)vec(X). (ii) Let A, B, D be n x n matrices and In the n x n identity matrix. Use the result from (i) to prove that AX +XB = D can be written as (In ® A) + (B" ® In))vec(X) = vec(D).

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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With each m xn matrix Y we associate the column vector vec(Y) of length m xn defined by vec(Y) := (y11,..., Yml, Y12, . .., Ym2, ..., Y1n,..., Ymn)“ . (i) Let A be an m x n matrix, B an p x q matrix, and C an m x q matrix. Let X be an unknown n x p matrix. Show that the matrir equation AXB = C is equivalent to the system of qm equations in np unknowns given by (BT ® A)vec(X) = vec(C') that is, vec(AX B) = (BT © A)vec(X). (ii) Let A, B, D be n x n matrices and In the n x n identity matrix. Use the result from (i) to prove that AX + XB = D can be written as ((In O A) + (B" ® In))vec(X) = vec(D).
Transcribed Image Text:With each m xn matrix Y we associate the column vector vec(Y) of length m xn defined by vec(Y) := (y11,..., Yml, Y12, . .., Ym2, ..., Y1n,..., Ymn)“ . (i) Let A be an m x n matrix, B an p x q matrix, and C an m x q matrix. Let X be an unknown n x p matrix. Show that the matrir equation AXB = C is equivalent to the system of qm equations in np unknowns given by (BT ® A)vec(X) = vec(C') that is, vec(AX B) = (BT © A)vec(X). (ii) Let A, B, D be n x n matrices and In the n x n identity matrix. Use the result from (i) to prove that AX + XB = D can be written as ((In O A) + (B" ® In))vec(X) = vec(D).
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