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The algebra of matrices.

As we saw previously, a matrix is a rectangular array of elements of $\mathbb{K} $; we say that the matrix is of order $m \times n$ if it has m rows and n columns. The set of all matrices of order $m \times n$ with entries in $\mathbb{K} $ is denoted $\mathcal{M}_{m \times n}(\mathbb{K} )$.

The matrix $\begin{pmatrix}a_{11} & a_{12} &a_{13} & \dots & a_{1n} \\
a_{21} & a_{22} &a...
...ts &\dots & \dots \\
a_{m1} & a_{m2} &a_{m3} & \dots & a_{mn}
\end{pmatrix}$is denoted $(a_{ij})_{\begin{matrix}1 \leq i \leq m \\ 1 \leq j \leq n \end{matrix}}$, or (aij) if no confusion is possible.

If m=n, the matrix is a square matrix of order n.

Equality:

Let A=(aij) and B=(bij) be two matrices in $\mathcal{M}_{m \times n}(\mathbb{K} )$.

$A=B \Longleftrightarrow \forall i,j, \; a_{ij}=b_{ij}$.

Addition:

Let A=(aij), B=(bij) and C=(cij) be three matrices in $\mathcal{M}_{m \times n}(\mathbb{K} )$.

$C=A+B \Longleftrightarrow \forall i,j, \; c_{ij}=a_{ij}+b_{ij}.$

The zero matrix: the zero matrix in $\mathcal{M}_{m \times n}(\mathbb{K} )$ is the matrix (aij)such that $\forall i,j, \quad a_{ij}=0$. Multiplication of a matrix by a scalar: Let A=(aij) and B=(bij) be two matrices in $\mathcal{M}_{m \times n}(\mathbb{K} )$ and take $\alpha \in \mathbb{K} $.

$A= \alpha B \Longleftrightarrow \forall i=1, \dots,m, \quad \forall j=1, \dots, n, \quad
a_{ij}= \alpha b_{ij}$.
Multiplication:

Let $A=(a_{ij}) \in \mathcal{M}_{m \times n}(\mathbb{K} )$, $B=(b_{ij}) \in \mathcal{M}_{n \times p}(\mathbb{K} )$ and $C=(c_{ij}) \in \mathcal{M}_{m \times p}(\mathbb{K} )$.

$C=A \cdot B \quad \Longleftrightarrow \quad \forall i, (1 \leq i \leq m), \fora...
...1 \leq j \leq p),
\quad c_{ij}= \underset{k=1}{\overset{n}{\sum}} a_{ik}b_{kj}$.

This definition is motivated by the link between matrices and linear operators: the product of two matrices corresponds to the composition of two linear operators (v.s.  1.4).



 
next up previous contents
Next: How to compute the Up: No Title Previous: Linear transformations and matrices.
Noah Dana-Picard
2001-02-26