Tuesday, June 26, 2007

Transpose of a matrix

In today's blog, I outline the properties of the tranpose of a matrix.

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) =

= Ent
j,i(A) + Entj,i(B) = Enti,j(AT) + Enti,j(AT) =

= Ent
i,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-1A)T = IT = I

(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

5 comments:

Anonymous said...

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) =

???

Laine said...

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TGMN said...

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Siddharth Rai said...

thanks!

kamal rawat said...

Thanks