Zero Matrix is Identity for Matrix Entrywise Addition
Theorem
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.
Then $\mathbf 0$ is the identity element for matrix entrywise addition.
Proof 1
From:
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings whose zero is the number $0$ (zero).
Hence we can apply Zero Matrix is Identity for Matrix Entrywise Addition over Ring.
$\Box$
The above cannot be applied to the natural numbers $\N$, as they do not form a ring.
However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid whose identity is $0$ (zero).
By definition, matrix entrywise addition is the Hadamard product with respect to addition of numbers.
The result follows from Zero Matrix is Identity for Hadamard Product.
$\blacksquare$
Proof 2
Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.
Then:
| \(\ds \mathbf A + \mathbf 0\) | \(=\) | \(\ds \sqbrk a_{m n} + \sqbrk 0_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqbrk {a + 0}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Identity Element of Addition on Numbers | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \mathbf A + \mathbf 0\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix |
Similarly:
| \(\ds \mathbf 0 + \mathbf A\) | \(=\) | \(\ds \sqbrk 0_{m n} + \sqbrk a_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqbrk {0 + a}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Identity Element of Addition on Numbers | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \mathbf 0 + \mathbf A\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix |
$\blacksquare$
Also see
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices: $3$