Today's content is taken from Matrices and Linear Transformations by Charles G. Cullen.
Definition 1: Enti,j(A)
Enti,j(A) refers to the entry in the (i,j) position of the matrix A. That is, on the ith row and the jth column.
Definition 2: Transpose of a matrix
The transpose of a matrix A which is denoted as AT is the matrix defined in the following way:
Enti,j(AT) = Entj,i(A)
The following are the properties of the transpose of a matrix:
Lemma 1: (AT)T = A
Proof:
For any entry of A:
Enti,j[(AT)T] = Entj,i(AT) = Enti,j(A)
QED
Lemma 2: If A,B are the same size, (A + B)T = AT + BT
Proof:
(1) For any entry of A:
Enti,j[(A + B)T] = Entj,i(A + B) =
= Entj,i(A) + Entj,i(B) = Enti,j(AT) + Enti,j(AT) =
= Enti,j(AT + BT)
(2) Thus,
(A + B)T = AT + BT
QED
Lemma 3: If AB is defined, then (AB)T = BTAT
Proof:
(1) Let A be an m x n matrix and B be an n x s matrix.
(2) Then, AB is an m x s matrix.
(3) Then (AB)T is an s x m matrix.
(4) BT is an s x n matrix and AT is an n x m matrix.
(5) So, BTAT is an s x m matrix.
(6) For any entry of (AB)T, we have:
Enti,j[(AB)T] = Entj,i(AB) =
= ∑ (k=1,n) Entj,k(A)*Entk,i(B) =
= ∑ (k=1,n) Entk,j(AT)*Enti,k(BT) =
= ∑ (k=1,n) Enti,k(BT)Entk,j(AT) =
= Enti,j(BTAT)
QED
Lemma 4: (aB)T = aBT for any scalar a
Proof:
(1) For any entry of (aB)T
Enti,j[(aB)T] = Entj,i(aB) = a*Entj,i(B) = a*Enti,j(BT)
(2) This gives us that:
(aB)T = aBT
QED
Lemma 5: If A is invertible, then so is AT and (AT)-1 = (A-1)T
Proof:
(1) Assume that A is invertible.
(2) AT(A-1)T =
= (A-1A)T [Using Lemma 3 above]
(3) (A
(4) (A-1)TAT = (AA-1)T [Using Lemma 3 above]
(5) (AA-1)T = IT = I
(6) Therefore, AT is invertible with (A-1)T as its inverse.
(7) So, we have that (A-1)T)AT = I.
(8) So, from step #6, we have that (AT)-1 = (A-1)T since (A-1)T is the inverse of AT.
QED
References
- Charles G. Cullen, Matrices and Linear Transformations, Dover Publications, Inc., 1972.
4 comments :
In the proof of Lemma 2, shouldn't
"""= Entj,i(A) + Entj,i(B) = Enti,j(AT) + Enti,j(AT) ="""
this part be written as:
= Entj,i(A) + Entj,i(B) = Enti,j(AT) + Enti,j(BT) =
???
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Thanks
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