Invertible matrix/Field/Similar/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$. Then ${}M$ is called invertible, if there exists a matrix ${}A\in \operatorname {Mat} _{n}(K)$ such that

${}A\circ M=E_{n}=M\circ A\,$

holds.

Definition

Let ${}K$ denote a field. For an invertible matrix ${}M\in \operatorname {Mat} _{n}(K)$, the matrix ${}A\in \operatorname {Mat} _{n}(K)$ fulfilling

${}A\circ M=E_{n}=M\circ A\,,$

is called the inverse matrix of ${}M$. It is denoted by

$M^{-1}.$

The product of invertible matrices is again invertible. Due to fact, the matrix describing a base change is invertible, and the matrix of the reversed base change is its inverse matrix.

Definition

For a field ${}K$ and ${}n\in \mathbb {N} _{+}$, the set of all invertible ${}n\times n$-matrices with entries in ${}K$ is called the general linear group

over ${}K$. It is denoted by ${}\operatorname {GL} _{n}\!{\left(K\right)}$.

Definition

Two square matrices ${}M,N\in \operatorname {Mat} _{n}(K)$ are called similar, if there exists an invertible matrix ${}B$ with

${}M=BNB^{-1}$.

For a linear mapping ${}\varphi \colon V\rightarrow V$, the describing matrices with respect to two bases are similar to each other, due to fact.