Matrix Transpose

The transpose of a matrix is denoted by is obtained by changing rows into columns or columns to rows of a matrix . If size of the matrix is then the size of the transposed matrix is .

Transpose Of A Matrix

The element in row and column of matrix becomes the row and column element in matrix .

\begin{aligned}
&(A^T)_{ij} = (A)_{ji}
\end{aligned}

Let be a matrix of size .

\begin{aligned}
A = \begin{bmatrix}a & b\\ c & d\\ e & f\end{bmatrix}
\end{aligned}

Transpose of matrix .

\begin{aligned}
A^T = \begin{bmatrix}a & c & e\\ b & d & f\end{bmatrix}
\end{aligned}

Let us take element ‘c’ which is at 2nd row and 1st column of matrix ; after transpose operation on matrix , it is at the position of 1st row and 2nd column of matrix .

Similarly, the element ‘b’ is at the position of first row and second column of matrix , but after the transpose operation, its position changes to 2nd row and 1st column in matrix .

Example #1

Transpose the following matrix A.

\begin{aligned}
A = \begin{bmatrix}3 & 1 & 5\\ 2 & 6 & 9\end{bmatrix}
\end{aligned}

The transpose of matrix is

\begin{aligned}
A^T = \begin{bmatrix}3 & 2\\ 1 & 6\\5 & 9\end{bmatrix}
\end{aligned}

Example #2

Transpose the following matrix B.

\begin{aligned}
&A = \begin{bmatrix}1 & 5\\ 7 & 6\\8 & 4\end{bmatrix}
\end{aligned}

The transpose of matrix is

\begin{aligned}
A^T = \begin{bmatrix}1 & 7 & 8\\ 5 & 6 & 4\end{bmatrix}
\end{aligned}

Symmetric Matrix

When the transpose of the matrix is the original matrix itself, then it is called a Symmetric matrix. Suppose is a matrix of size , then the transpose of matrix .

All the elements above the diagonal is a mirror image of elements below the diagonal elements. That is, is symmetric matrix if for all i and j.

\begin{aligned}
A = \begin{bmatrix}a_{11} & p & q\\p & a_{22} & r\\q & r & a_{33}\end{bmatrix}
\end{aligned}

The elements of . The transpose of such a matrix is,

\begin{aligned}
A^T = \begin{bmatrix}a_{11} & p & q\\p & a_{22} & r\\q & r & a_{33}\end{bmatrix}
\end{aligned}

Therefore,

\begin{aligned}
A = A^T
\end{aligned}

What Are The Properties Of A Transpose Of A Matrix ?

In this section, we shall discuss about the properties of a transpose of a matrix. There are 4 interesting properties of a transpose as listed below.

1. , where is a matrix of size or .
2. , where and are of same size, that is, or .
3. , where is matrix of size or and is a real number.
4. , where and are matrices of size and .

Let us verify each of the statement.

#1 :

The transpose of a transpose of matrix is the original matrix .

\begin{aligned}
Let \hspace{5px}A = \begin{bmatrix}2 & 3\\-1 & 5\end{bmatrix}
\end{aligned}

Transpose of matrix .

\begin{aligned}
A^T = \begin{bmatrix}2 & -1\\3 & 5\end{bmatrix}
\end{aligned}

Transpose of .

\begin{aligned}
(A^T)^T = \begin{bmatrix}2 & 3\\-1 & 5\end{bmatrix}
\end{aligned}

From the results above, it is clear that where is a matrix of size or .

#2 :

The transpose of sum of two matrices and of same size or is equal to sum of transpose of matrices and .

Let and be two matrices of same size. Then

\begin{aligned}
&A = \begin{bmatrix}1 & 5\\-2 & 3\end{bmatrix}   B = \begin{bmatrix}2 & 1\\5 & -1\end{bmatrix}\\\\
&(A + B) = \begin{bmatrix}3 & 6\\3 & 2\end{bmatrix}
\end{aligned}

Transpose of matrix .

\begin{aligned}
(A + B)^T = \begin{bmatrix}3 & 3\\6 & 2\end{bmatrix}
\end{aligned}

Now, we shall take transpose of matrix and matrix and add them together to obtain .

\begin{aligned}
A = \begin{bmatrix}1 & 5\\-2 & 3\end{bmatrix}   B = \begin{bmatrix}2 & 1\\5 & -1\end{bmatrix}
\end{aligned}

Transpose of A.

\begin{aligned}
A^T = \begin{bmatrix}1 & -2\\5 & 3\end{bmatrix}
\end{aligned}

Transpose of B.

\begin{aligned}
B^T = \begin{bmatrix}2 & 5\\1 & -1\end{bmatrix}
\end{aligned}

Sum of and .

\begin{aligned}
A^T + B^T = \begin{bmatrix}3 & 3\\6 & 2\end{bmatrix}
\end{aligned}

#3 :

A transpose of the product of matrix with scalar is equal to the product of scalar and transpose of matrix where size of the matrix is or and is a real number.

\begin{aligned}
&Let \hspace{5px} A = \begin{bmatrix}2 & 3\\1 & 7\end{bmatrix} \hspace{4px} and \hspace{5px} r = 2\\\\
&(rA) = \begin{bmatrix}4 & 6\\2 & 14\end{bmatrix}
\end{aligned}

Transpose of ,

\begin{aligned}
(rA)^T = \begin{bmatrix}4 & 2\\6 & 14\end{bmatrix}
\end{aligned}

Similarly, let us take transpose of .

\begin{aligned}
A^T = \begin{bmatrix}2 & 1\\3 & 7\end{bmatrix}
\end{aligned}

The product is,

\begin{aligned}
rA^T = \begin{bmatrix}4 & 2\\6 & 14\end{bmatrix}
\end{aligned}

Therefore, .

The output of both the products are equal and the property is true for all matrices.

#4 :

The transpose of product of two defined ( and ) matrices and is equal to the product of transpose of matrix and transpose of matrix . Let us verify this claim with the help of an example.

\begin{aligned}
&Let \hspace{5px}A = \begin{bmatrix}1 & 5\\2 & 1\end{bmatrix} \hspace{5px} and \hspace{5px} B = \begin{bmatrix}3 & -1\\2 & 3\end{bmatrix}\\\\
&AB = \begin{bmatrix}3 + 10 & -1 + 15\\6 + 2 & -2 + 3\end{bmatrix}= \begin{bmatrix}13 & 14\\8 & 1\end{bmatrix}
\end{aligned}

Transpose of .

\begin{aligned}
(AB)^T = \begin{bmatrix}13 & 8\\14 & 1\end{bmatrix}
\end{aligned}

Similarly, the transpose of matrix and matrix is,

\begin{aligned}
&B^T = \begin{bmatrix}3 & 2\\-1 & 3\end{bmatrix} \hspace{5px} and \hspace{5px} A^T = \begin{bmatrix}1 & 2\\5 & 1\end{bmatrix}\\\\
&B^TA^T = \begin{bmatrix}3 + 10 & 6 + 2\\-1 + 15 & -2 + 3\end{bmatrix}\\\\
&B^TA^T = \begin{bmatrix}13 & 8\\14 & 1\end{bmatrix}\\\\
&Therefore, \hspace{5px}(AB)^T = B^TA^T
\end{aligned}

Once again, the product of both sides of the equation of the property holds true. The property is valid.

In the next, post we will discuss more about symmetric and skew-symmetric matrices.