In this explainer, we will learn how to find the transpose of a matrix and identify symmetric and skew-symmetric matrices.
Let us first recall the definition of a matrix.
An matrix is a rectangular array of entries with rows and columns, defined by where and . Each is called an entry or an element of the matrix.
Now, we want to investigate taking the transpose of a matrix, which is an operation that flips a matrix over its diagonal entries. To best illustrate this concept, let us consider the following matrix.
As per our definition above, this can be written as , where and . When we refer to the diagonal entries, we mean entries of the form , which, in this case, are the entries and . We highlight these as follows:
Now, to take the transpose of a matrix, we flip the matrix over the diagonal so the rows become the columns and the columns become the rows. The transposed matrix is denoted by , where the superscript stands for “transposed.” Let us show this below:
Notice how only the diagonal entries remain in the same place, while entries below the diagonal have been swapped with those above the diagonal. Specifically, the rows and columns have traded places. To visualize this, let us highlight the first row of the original matrix:
In the transposed matrix, this becomes the first column:
Similarly, the second and third rows become the second and third columns, as shown below:
So, as we can see, matrix transposition can be thought of as swapping the rows and the columns with each other.
Since the rows and the columns of a matrix are swapped when we transpose a matrix, the number of rows and the number of columns of a matrix are also swapped. For matrix , which has order , we can see that has order . We can generalize this in the following way.
Definition: Order of the Transpose of a Matrix
If a matrix is of order , then the transpose of the matrix is of order .
In our first example, we will consider a direct use of this definition to find the order of a matrix.
Example 1: Finding the Order of a Matrix Transpose
If is a matrix of order , then what is the order of the matrix ?
Matrix is the transpose of matrix . We recall that the transpose of a matrix swaps its rows with its columns. In particular, this means that the number of rows and the number of columns of the matrix will be interchanged. In other words, if a matrix is of order , the transpose of the matrix will be of order .
In this example, we are given that matrix has order , which means and . Hence, the transpose of , which is , is of order
Visually, we can see that the transpose has the following effect on the order of the matrix:
Hence, the order of the matrix is .
In the previous example, we found the order of the transpose of a matrix. Let us now discuss the full definition of the transpose of a matrix and how to identify its entries.
Definition: Transpose of a Matrix
Consider an matrix defined by . The transpose of , denoted , is an matrix defined by
In the above definition, we see that the row and column indices ( and ) are swapped around. This has the same effect of swapping the rows and the columns of the matrix, as we have seen above. For instance, we can see below that entry in matrix becomes entry in matrix :
In other words, the entry in row 3 and column 1 becomes the entry in row 1 and column 3. Any diagonal entries, such as , do not change.
When we want to find the transpose of a matrix, the first step is to find the order of the new matrix. After this, we have to find the entries by flipping the matrix in the diagonal. In the next example, we will practice how to do this.
Example 2: Finding the Transpose of a Matrix
Given that find .
We know that is the transpose of matrix , so we need to find the transpose of the given matrix in this example.
Recall that the transpose of an matrix is of order . Since is a matrix, the transpose of will be a matrix, leading to the following matrix with blank entries:
Now, let us fill in the entries of matrix . Recall that the transpose of a matrix swaps its rows with its columns. In other words, the first row becomes the first column, the second row becomes the second column, and so on. Thus, let us consider each row of and write each as corresponding columns of .
The first row of is
In the transposed matrix, this becomes the first column:
Next, we have the second row:
This becomes the second column:
This leads to the transposed matrix
In the previous example, we found the transpose of a given matrix by placing each row of the matrix as the corresponding column of the transposed matrix. This method can be applied to matrices of any order, as we explain below.
How To: Transposing a Matrix
To find the transpose of an matrix, we can follow this procedure:
- Identify that the order of the transpose matrix is and that it thus has rows and columns as shown below:
- Consider each row of the matrix one by one and write it as a corresponding column in the transposed matrix:
- Complete the process to get the transpose of :
It should be noted that the matrix transpose has a special property when it is applied twice in succession. We will observe this property through the next example.
Example 3: Applying the Matrix Transpose Twice in Succession
Given the matrix find .
We know that a superscript next to a matrix indicates the transpose of a matrix. Here, we see that there are two superscripts. To understand this notation, we first note that is the transpose of the given matrix . Then, we note that is the transpose of the transpose of the given matrix , meaning that we need to apply the transpose to matrix twice successively.
Therefore, in this example, we need to apply the transpose to matrix twice in succession. We can begin by first finding matrix and then applying the transpose to the resulting matrix.
Recall that the transpose of an matrix is of order . Since is a matrix, will be a matrix, leading to the following matrix with blank entries:
Recall that we can find the transpose of a matrix by swapping the rows and the columns around with each other. Hence, the first row of becomes the first column of :
Similarly, we write the second row of as the second column of :
Thus, we have obtained the transpose of :
We repeat the process by taking the transpose again in order to find . Since is a matrix, will be a matrix, leading to the following matrix with blank entries:
As we did before, let us swap the rows of around with the columns of . Starting with the first row, we have
The second row gives us
For the final row, we have
As we can see, this matrix is actually identical to . This is because swapping the rows and columns twice results in them being the same as they were before. In conclusion, we have
In the previous example, we saw that applying the transpose to a given matrix twice in succession resulted in the same matrix. In fact, this is a general rule that applies to any matrix.
Rule: Applying the Transpose to a Matrix Twice
For a matrix , applying the transpose twice results in the same matrix. In other words,
Having seen that successive applications of the transpose result in the same matrix, we might wonder whether there are any other properties that the matrix transpose has. For instance, are there any matrices such that if we apply the transpose just once, we still get the same matrix? As it turns out, there is a specific class of matrices that obey such a property.
Definition: Symmetric and Skew-Symmetric Matrices
Let be an matrix (i.e. a square matrix). Then, is a symmetric matrix if
Alternatively, is a skew-symmetric matrix if
We note that, in the above definition, the matrix has to be a square matrix. This is because, unless the matrix is square, the order of the matrix will change when we apply the transpose to the matrix. Hence, symmetric and skew-symmetric matrices must be square matrices.
Also, recall that, when we take the matrix transpose, we swap around the indices of each entry, so . Consequently, in order for a matrix to be equal to its transpose, we require that . Similarly for skew-symmetric matrices, we require . We note that a consequence of this is that the diagonal entries of a skew-symmetric matrix have to be zero, since would not be satisfied otherwise. Let us summarize these properties below.
Property: Indices of Symmetric and Skew-Symmetric Matrices
Let be a square matrix. is a symmetric matrix if and only if
Similarly, is a skew-symmetric matrix if and only if
With both the definition and the above property in mind, we can begin to imagine what a symmetric or skew-symmetric matrix looks like. Consider the following matrix, where the diagonal has been highlighted:
Notice that the matrix would remain the same if we reflect each entry over the diagonal. In this sense, the matrix is symmetric, where the axis of symmetry is the diagonal. In terms of indices, we can see that for every ; for instance, and . We note that, by default, the diagonal entries obey this property since they are of the form (swapping their row and column indices has no effect).
Skew-symmetric matrices can be identified in a similar manner. Consider the following matrix:
In this case, on either side of the diagonal, the entries are the same but with the opposite signs. For instance, , while , and , while . In particular, the diagonal entries are all zero, i.e., for . In general, we can see that ; thus, is skew symmetric. We can see that the diagonal entries being 0 is a necessary consequence of this since , which can only be true for .
As an additional note, let us consider the case where we have a square-zero matrix:
Taking the transpose of this matrix would have no effect, since all the entries are the same. Thus, and is symmetric. However, we also note that , which means that is skew symmetric too. In fact, square-zero matrices are the only type of matrix that are both symmetric and skew symmetric at the same time.
Let us now apply our knowledge of symmetric matrices in the following example.
Example 4: Finding the Unknown Elements That Make a Given Matrix Symmetric
Find the value of that makes the matrix symmetric.
Recall that, in order for a matrix to be symmetric, it must be square and it must satisfy where is the transpose of . As the number of rows and the number of columns of are both equal to 2, it is indeed a square matrix, satisfying the first condition. In order to find what values of satisfy the second condition, we can apply the transpose to and equate it to itself.
To begin, let us calculate . Since is a matrix, the order of the transposed matrix will also be . To find the entries of , we can rewrite the rows of as columns of as follows:
Thus, the transpose is
To satisfy , we set this matrix equal to the original matrix given to us to get the following matrix equation:
To check the equality of two matrices, we must check that the corresponding entries are equal. Clearly, the diagonal entries are the same, as the transpose does not change them. Let us highlight the off-diagonal entries that must be equal below:
This gives us two equations we must satisfy:
We can see that both equations are the same. As this is a linear equation, we can be solve it by rearranging it so that is the subject:
Note that, in the previous example, it would also be possible to find by using the following property of symmetric matrices:
For a matrix, we only need to verify that . We can see that in the given matrix:
Thus, setting leads to the same equation that we found in the example.
Let us now consider an example where we consider a skew-symmetric matrix.
Example 5: Finding the Unknown Elements That Make a Given Matrix Skew Symmetric
Given that the matrix is skew symmetric, find the value of .
Recall that, in order for a matrix to be skew symmetric, it must be square and it must satisfy the condition where is the transpose of . As the number of rows and the number of columns of are both equal to 3, this is indeed a square matrix. By applying the transpose to and equating it to itself, we can hopefully find what values of , , and satisfy the second condition.
Let us start by calculating . Since is of order , the order of will also be . To find the entries of , we can rewrite the rows of as columns of as follows:
Thus, the transpose is
We can now write using the negative of this transposed matrix and the original matrix:
In order for this matrix equation to be satisfied, the corresponding entries must be equal. We, first of all, note that on both sides of the equation, the diagonal entries are zero (which is a necessary condition for a matrix to be skew symmetric). For the off-diagonal entries, we highlight where they must be equal:
This results in six equations, although in the cases where we have used the same highlight color, the equations are equivalent. For instance, the green entries in positions and give us the following equivalent pair of equations:
Thus, we have three unique equations:
To calculate , let us first solve these simultaneous equations to find , and individually. From the third equation, we have
So, . We can substitute this into the first equation, , to get
Thus, . Finally, we can substitute this into the second equation, , to get
All in all, this gives us , , and . Finally, we can find by adding them together to get
In our final example, we will consider how the transpose interacts with other matrix operations.
Example 6: Investigating the Properties of Transpose Matrices
Given the matrices and , does ?
We know that a superscript next to a matrix indicates the transpose of a matrix. The left-hand side of the given equation applies the transpose after the matrix subtraction, while the right-hand side applies the transpose to each matrix before the matrix subtraction. Hence, this example is asking whether we can interchange the order of the transpose and the subtraction.
Now, to check whether this equation is correct, we need to calculate each side of it and verify that the matrices are equal.
To compute the left-hand side of the equation, let us first find the difference . Recall that we can take the difference of two matrices of the same order by taking the difference of the corresponding entries in the two matrices. This gives us
Then, we can apply the transpose to find the matrix . Recall that the transpose of an matrix is an matrix. Since is a matrix, its transpose must have the same order.
We also recall that we can find the transpose of a matrix by swapping its rows with its columns. We can take each row of matrix and place each as the corresponding column as follows:
So, we have
Now, for the right-hand side, we need to calculate . Let us begin by finding the transpose of and separately. Let us first consider :
We note that is a symmetric matrix, since the values on either side of the diagonal ( and ) are equal. Thus, the transpose is
Next, let us consider :
This is not a symmetric matrix, so we will take the transpose by swapping its rows with its columns:
Thus, we have
Finally, we can calculate using matrix subtraction as follows:
As we can see, this is equal to the matrix we calculated for . Thus, we can conclude that .
We note that the result of the example above is not coincidental. For general matrices and , it holds true, and it also extends to matrices under addition. We call this property distributivity.
Property: Distributivity of Matrix Transposition
Let and be two matrices. Therefore, we have
In this explainer, we have discussed the matrix transpose and some of its interesting properties. Let us summarize the key points we have learned.
- If a matrix has order , then the transpose of the matrix has order .
- If , then the transpose matrix is defined by
- We can transpose the entries of a matrix by writing its rows as columns.
- A square matrix is symmetric if Additionally, a square matrix is symmetric if and only if
- A square matrix is skew symmetric if Similarly, a square matrix is skew symmetric if and only if
- Let and be two matrices. Therefore, we have