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: Ent_i,j(A)

Ent_i,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 A^T is the matrix defined in the following way:

Ent_i,j(A^T) = Ent_j,i(A)

The following are the properties of the transpose of a matrix:

Lemma 1: (A^T)^T = A

Proof:

For any entry of A:

Ent_i,j[(A^T)^T] = Ent_j,i(A^T) = Ent_i,j(A)

QED

Lemma 2: If A,B are the same size, (A + B)^T = A^T + B^T

Proof:

(1) For any entry of A:

Ent_i,j[(A + B)^T] = Ent_j,i(A + B) =

= Ent_j,i(A) + Ent_j,i(B) = Ent_i,j(A^T) + Ent_i,j(A^T) =

= Ent_i,j(A^T + B^T)

(2) Thus,

(A + B)^T = A^T + B^T

QED

Lemma 3: If AB is defined, then (AB)^T = B^TA^T

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) B^T is an s x n matrix and A^T is an n x m matrix.

(5) So, B^TA^T is an s x m matrix.

(6) For any entry of (AB)^T, we have:

Ent_i,j[(AB)^T] = Ent_j,i(AB) =

= ∑ (k=1,n) Ent_j,k(A)*Ent_k,i(B) =

= ∑ (k=1,n) Ent_k,j(A^T)*Ent_i,k(B^T) =

= ∑ (k=1,n) Ent_i,k(B^T)Ent_k,j(A^T) =

= Ent_i,j(B^TA^T)

QED

Lemma 4: (aB)^T = aB^T for any scalar a

Proof:

(1) For any entry of (aB)^T

Ent_i,j[(aB)^T] = Ent_j,i(aB) = a*Ent_j,i(B) = a*Ent_i,j(B^T)

(2) This gives us that:

(aB)^T = aB^T

QED

Lemma 5: If A is invertible, then so is A^T and (A^T)^-1 = (A^-1)^T

Proof:

(1) Assume that A is invertible.

(2) A^T(A^-1)^T =

= (A^-1A)^T [Using Lemma 3 above]

(3) (A-1A)^T = I^T = I

(4) (A^-1)^TA^T = (AA^-1)^T [Using Lemma 3 above]

(5) (AA^-1)^T = I^T = I

(6) Therefore, A^T is invertible with (A^-1)^T as its inverse.

(7) So, we have that (A^-1)^T)A^T = I.

(8) So, from step #6, we have that (A^T)^-1 = (A^-1)^T since (A^-1)^T is the inverse of A^T.

QED

References

• Charles G. Cullen, Matrices and Linear Transformations, Dover Publications, Inc., 1972.