The transpose of the identity matrix is the identity matrix itself.
Understanding the Identity Matrix
An identity matrix, often denoted as $I$ or $I_n$ (where $n$ represents its dimensions), is a special type of square matrix. It features ones along its main diagonal (from the top-left to the bottom-right) and zeros everywhere else. Its primary role in linear algebra is analogous to the number '1' in scalar multiplication: when multiplied by another matrix, it leaves that matrix unchanged.
For example, a 3x3 identity matrix looks like this:
| Row/Column | Column 1 | Column 2 | Column 3 |
|---|---|---|---|
| Row 1 | 1 | 0 | 0 |
| Row 2 | 0 | 1 | 0 |
| Row 3 | 0 | 0 | 1 |
The Concept of a Matrix Transpose
In linear algebra, the transpose of a matrix is an operation that effectively flips the matrix over its main diagonal. This process involves switching the row and column indices of each element. If a matrix is denoted as $A$, its transpose is typically represented as $A^T$ or $A'$.
To find the transpose:
- The element at row $i$, column $j$ in the original matrix becomes the element at row $j$, column $i$ in the transposed matrix.
- The first row of the original matrix becomes the first column of the transposed matrix, the second row becomes the second column, and so on.
Why the Identity Matrix is Its Own Transpose
The identity matrix possesses a unique property that makes it symmetric: all its elements are already arranged symmetrically with respect to its main diagonal.
Consider the elements of an identity matrix $I$:
- If an element is on the main diagonal (i.e., its row index equals its column index, $i=j$), it is a '1'.
- If an element is not on the main diagonal (i.e., its row index does not equal its column index, $i \neq j$), it is a '0'.
When you apply the transpose operation, you switch the row and column indices.
- For diagonal elements ($I{ii}$), switching $i$ and $i$ results in the same position, so $I{ii}^T = I_{ii}$. These values remain '1'.
- For off-diagonal elements ($I{ij}$ where $i \neq j$), the value is '0'. When transposed to $I{ji}^T$, this position also originally held a '0'.
Since flipping the identity matrix over its diagonal does not change any of its elements or their positions relative to the diagonal, the resulting matrix is identical to the original identity matrix. This property makes the identity matrix an example of a symmetric matrix, where $A = A^T$.
Example: Transpose of a 2x2 Identity Matrix
Let's take a 2x2 identity matrix, $I_2$:
$I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$
To find its transpose, $I_2^T$:
- The element in row 1, column 1 (which is 1) remains in row 1, column 1.
- The element in row 1, column 2 (which is 0) moves to row 2, column 1.
- The element in row 2, column 1 (which is 0) moves to row 1, column 2.
- The element in row 2, column 2 (which is 1) remains in row 2, column 2.
$I_2^T = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$
As seen, $I_2^T$ is exactly the same as $I_2$. This demonstrates that the transpose of an identity matrix is the identity matrix itself, regardless of its size.
