Transpose of a Matrix
Another important operation on matrices is that of taking the transpose. For a matrix , we denote the transpose of by . Before formally defining the transpose, we explore this operation on the following matrix. \begin{equation*} \begin{bmatrix} 1 & 4 \\ 3 & 1 \\ 2 & 6 \end{bmatrix}^{T}= \begin{bmatrix} 1 & 3 & 2 \\ 4 & 1 & 6 \end{bmatrix} \end{equation*}
What happened? The first column became the first row and the second column became the second row. Thus the matrix became a matrix. The number was in the first row and the second column and it ended up in the second row and first column.
The definition of the transpose is as follows.
The Transpose of a Matrix Let be an matrix. Then the transpose of , denoted by , is
the matrix given by \begin{equation*} A^{T} = \begin{bmatrix} a _{ij}\end{bmatrix}^{T}= \begin{bmatrix} a_{ji} \end{bmatrix} \end{equation*}
The -entry of becomes the -entry of .
Calculate for the following matrix \begin{equation*} A = \begin{bmatrix} 1 & 2 & -6 \\ 3 & 5 & 4 \end{bmatrix} \end{equation*}
We will prove Property item:matrixtranspose1. The remaining properties are left as exercises.
- Proof of Property item:matrixtranspose1:
- Note that and have compatible dimensions, so that
is defined and has dimensions . Thus, has dimensions . On the right side of
the equality, has dimensions , and has dimensions . Therefore is defined and
has dimensions . Now we know that and have the same dimensions.
To show that we need to show that their corresponding entries are equal. Recall that the -entry of is given by the dot product of the row of and the column of . The same dot product is also the -entry of .
The -entry of is given by the dot product of the row of and the column of . But the row of is has the same entries as the column of , and the column of has the same entries as the row of . Therefore the -entry of is also equal to the -entry of .
Thus, the corresponding components of are equal and we conclude that .
The transpose of a matrix is related to other important topics. Consider the following definition.
Symmetric and Skew Symmetric Matrices An matrix is said to be symmetric if It is
said to be skew symmetric if
We will explore these definitions in the following examples.
Let \begin{equation*} A= \begin{bmatrix} 2 & 1 & 3 \\ 1 & 5 & -3 \\ 3 & -3 & 7 \end{bmatrix} \end{equation*}
Show that is symmetric.
Let \begin{equation*} A= \begin{bmatrix} 0 & 1 & 3 \\ -1 & 0 & 2 \\ -3 & -2 & 0 \end{bmatrix} \end{equation*}
Show that is skew symmetric.
\begin{equation*} A^{T} = \begin{bmatrix} 0 & -1 & -3\\ 1 & 0 & -2\\ 3 & 2 & 0 \end{bmatrix} \end{equation*}
Each entry of is equal to times the same entry of . Hence, and so by Definition def:symmetricandskewsymmetric, is skew symmetric.
A special case of a symmetric matrix is a diagonal matrix. A diagonal matrix is a square matrix whose entries outside of the main diagonal are all zero. The identity matrix is a diagonal matrix. Here is another example.
Practice Problems
Prove Properties item:transoftrans, item:matrixtranspose2 and item:matrixtranspose3 of Theorem th:transposeproperties.
Problems prob:symmetricclassification1-prob:symmetricclassification3
Classify each matrix as symmetric, skew symmetric, or neither.
Make a conjecture about the main diagonal entries of a skew symmetric matrix.
Prove your conjecture.
Text Source
The text in this section is an adaptation of Section 2.1 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, p. 68-70.
2024-09-26 20:46:16