In this explainer, we will learn how to find the transpose of a matrix, elements of a given row and column after transposing, and a matrix’s dimensions after transposing.
The transpose of a matrix is a relatively new concept in linear algebra. The main ideas of this field were developed over several millennia, arguably beginning around the years 300–200 BC as a way of solving systems of linear equations. A better, more complete understanding of linear algebra was developed in the late 1600s, principally by Leibniz and Lagrange, with the introduction of essential concepts such as the determinant. The eminent mathematician Gauss worked intensively on linear algebra in the early 1800s, eventually coauthoring the powerful Gauss-Jordan elimination algorithm to solve systems of linear equations.
Given the range of sophisticated concepts that drove the embryonic study of linear algebra, it is perhaps surprising that a relatively simple concept—the matrix transpose—was not defined until 1858 by Cayley, by which point many key pillars of linear algebra had already been constructed and well understood. Despite this development occurring relatively late, the matrix transpose was so important as a concept that it forms the basis of many theorems and results that are studied by all students of linear algebra.
We will begin by defining the matrix transpose and will then illustrate this concept with an example, before completing some more problems.
Definition: Matrix Transpose
Consider a matrix that is specified by the formula
The matrix “transpose” is then a matrix that is composed of the elements of by the formula
As ever in linear algebra, the definition of this particular concept is not completely clear until it has been demonstrated by examples. Transposing a matrix has the effect of “flipping” the matrix along the diagonal entries. An alternative way of viewing this operation is that the transpose of switches the rows with the columns. Therefore, if has rows and columns, then the transpose will have rows and columns. Again, this is easier to demonstrate than it is to describe, so we will now provide an illustrative example.
Consider the matrix
This matrix has 3 rows and 2 columns and therefore the transpose will have 2 rows and 3 columns, hence having the form where the symbols represent values that are yet to be calculated.
Now we populate by taking the first row of , and writing the elements in the same order but now as the first column of :
We then write the second row of , as the second column of :
Finally, we write the third row of , as the third column of : hence completing the matrix transpose.
Note that if we write and next to each other and highlight only the diagonal entries, as below, then we observe that the diagonal entries are unchanged. This is true whenever we take the transpose of a matrix and the reason why we often simply refer to the transpose of a matrix as “flipping” along the diagonal entries.
To explain this, we refer to the definition above. For a matrix , the transpose is calculated using the same entries but referring to the row position as the column position and vice versa, which is encapsulated by the expression . For example, the entry refers to the entry in the second row and the first column of . Switching the index and the index gives , which corresponds to the entry in the first row and second column. This can be observed for the matrices and above.
However, the diagonal entries are where the row and column number are the same, meaning that , giving the entries . Even if the row index and the column index are switched, the result is the same entry . Therefore, all diagonal entries are unchanged by transposition, which is a key guiding result when computing the transpose of a matrix.
Example 1: Finding the Transpose of a Matrix
Find the transpose of the matrix .
We label this matrix as . This has 2 rows and 3 columns, which means that will have 3 rows and 2 columns. Therefore, will take the form where the represent entries that must be found.
Since the diagonal entries are unchanged when transposing a matrix, we highlight these in the original matrix, and copy them into the transpose matrix, as shown:
We then highlight the first row in the original matrix, and write these as the first column of the transpose matrix:
Then, we highlight the second row of and write these entries in order as the second column of the transpose matrix
Given that the matrix transpose is usually straightforward to calculate, it is unlikely that this operation would be interesting unless it had either some special algebraic properties or some useful applications. As luck would have it, the matrix transpose has both. As well as being useful in the definition of symmetric and skew-symmetric matrices (both of which are highly important concepts), the matrix transpose is endowed with a range of convenient algebraic properties, one of which is as follows.
Theorem: Double Application of the Matrix Transpose
For a matrix , applying the matrix transpose twice returns the original matrix. In other words,
This may be obvious, given that the transpose of a matrix would flip it along the diagonal entries and then applying the transpose again would simply flip it back. Using the alternative understanding, the matrix transpose would switch the rows and columns and applying this action again would switch them back. However, to properly illustrate that this is indeed the case, we consider the following example.
Example 2: The Transpose of the Transpose
Given the matrix, find .
The matrix has 2 rows and 3 columns and so the matrix will have 3 rows and 2 columns:
Knowing that the diagonal entries are unchanged, we immediately populate these entries in :
The first row of then becomes the first column of :
Then, the second row of becomes the second column of :
Now we wish to find the transpose of , which we denote . Due to having 3 rows and 2 columns, the transpose will have 2 rows and 3 columns:
We can identify that and have the same number of rows and columns, which is encouraging since otherwise there would have been no possibility of the two matrices being equal. As before, we first populate the diagonal entries of the unknown matrix:
Now we rewrite the first row of the left-hand matrix as the first column of the right-hand matrix:
The same process is then applied for the second row and the second column:
Finally, we write the entries in the third row as the entries of the third column:
We have therefore shown, in this example, that .
We could have equally proven this result with reference to the definition that
Given that taking the transpose switches the row index with the column index, we would find that thus showing that .
As we have already discussed, transposing a matrix once has the effect of switching the number of rows and columns. In other words, if has rows and columns, then will have rows and columns. This result can alternatively be summarized by the following theorem and example.
Theorem: Matrix Order and Transpose
If a matrix has order , then has order .
A consequence of this theorem is that if is a square matrix then will also be a square matrix of the same order. This can be easily shown by specifying that must have the same number of rows and columns, hence making it a square matrix with an order of . Even if we switch the rows for the columns in the transpose matrix, there will still be the same number of rows and columns, meaning that will also be a matrix of order and hence will be a square matrix of the same dimension as the original matrix .
Example 3: The Order of a Transpose Matrix
If is a matrix of order , then what is the order of the matrix ?
For a matrix with order , the transpose of the matrix has order . Following this result, if has order then the transpose is a matrix of order .
Now that we are more familiar with calculating the transpose of a matrix, we will solve two problems featuring this idea. Note that, in the following problems, the transpose of a matrix appears as part of a series of other algebraic operations involving matrices, which is very often the case when working in linear algebra.
Example 4: Equations Involving Matrix Transposition
Given that determine the value of .
The order of is , meaning that this is a square matrix. Therefore, will also be a square matrix of order . Given that we find that
We are asked to calculate , which gives
It can be observed that the matrix is equal to the negative of its own transpose, which is represented algebraically as . This means that is in fact a “skew-symmetric” matrix which is an important type of matrix that is defined with reference to the matrix transpose. It is the case with all skew-symmetric matrices that , which is validated in the matrix above, where we find that .
Example 5: Transposition and Matrix Subtraction
Given the matrices does ?
First, we calculate
For the right-hand side of the given equation, we first observe that is equal to its own transpose (meaning that this is a “symmetric” matrix). We can therefore write and hence simplify the following calculation:
We have therefore shown for this example that .
The example above actually points towards a much more general result which relates together the operation of transposition and the operations of addition and subtraction. To demonstrate this result, we define the matrices
We first choose to calculate
Next, we calculate
It is the case in this example that . Had we wished to, we could also have shown that . These two results are not accidental and can be summarized by the following theorem.
Theorem: Matrix Transpose and Addition/Subtraction
If and are two matrices of the same order, then
We would say that matrix transposition is “distributive” with respect to addition and subtraction.
There are many other key properties of matrix transposition that are defined in reference to other concepts in linear algebra, such as the determinant, matrix multiplication, and matrix inverses. When working in linear algebra, knowledge of the matrix transpose is therefore a vital and robust part of any mathematician’s tool kit.
- For a matrix defined as , the transpose matrix is defined as .
- In practical terms, the matrix transpose is usually thought of as either (a) flipping along the diagonal entries or (b) “switching” the rows for columns.
- A double application of the matrix transpose achieves no change overall. In other words, .
- If the order of is , then the order of is .
- The matrix transpose is “distributive” with respect to matrix addition and subtraction, being summarized by the formula .