Matrices/Several concepts/Section

Definition

The ${}n\times n$-matrix

${}E_{n}:={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,$

is called identity matrix.

The identity matrix ${}E_{n}$ has the property ${}E_{n}M=M=ME_{n}$, for an arbitrary ${}n\times n$-matrix ${}M$. Hence, the identity matrix is the neutral element with respect to matrix multiplication.

Remark

If we multiply an ${}m\times n$-matrix ${}A=(a_{ij})_{ij}$ with a column vector ${}x={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}$, then we get

${}Ax={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}\end{pmatrix}}\,.$

Hence, an inhomogeneous system of linear equations with disturbance vector ${}{\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}$ can be written briefly as

${}Ax=c\,.$

Then, the manipulations on the equations that do not change the solution set, can be replaced by corresponding manipulations on the rows of the matrix. It is not necessary to write down the variables.

Definition

An ${}n\times n$-matrix of the form

${\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}$

is called a diagonal matrix.

Definition

Let ${}K$ be a field, and let ${}M=(a_{ij})_{ij}$ be an ${}m\times n$-matrix over ${}K$. Then the ${}n\times m$-matrix

${M^{\text{tr}}}={\left(b_{ij}\right)}_{ij}{\text{ with }}b_{ij}:=a_{ji}$

is called the transposed matrix of ${}M$.

The transposed matrix arises by interchanging the roles of the rows and the columns. For example, we have

${}{{\begin{pmatrix}t&n&o&d\\r&s&s&x\\a&p&e&y\end{pmatrix}}^{\text{tr}}}={\begin{pmatrix}t&r&a\\n&s&p\\o&s&e\\d&x&y\end{pmatrix}}\,.$