Identity Matrix is Permutation Matrix
Jump to navigation
Jump to search
Theorem
An identity matrix is an example of a permutation matrix.
Proof
An identity matrix, by definition, has instances of $1$ on the main diagonal and $0$ elsewhere.
Each diagonal element is by definition on one row and one column of the matrix.
Also by definition, each diagonal element is on a different row and column from each other diagonal element.
The result follows by definition of permutation matrix.
$\blacksquare$
Sources
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.3$: $m \times n$ matrices