> [!Definition] Definition (Square matrix) > For any $n\in\mathbb{N}$, any [[Matrix|matrix]] in $\mathbb{R}^{n\times n}$ is a square (i.e number of rows equals number of columns). > # Properties **Algebra**: By [[Real Square Matrices Form Ring With Unity]], ... It's [[Unit Group of Ring|unit group]] is given by [[General Linear Group]]. > [!NOTE] Lemma (Linear Operator) > Any a square matrix has a corresponding [[Linear maps|linear operator]] $\mathbb{R}^{n}\to \mathbb{R}^{n}$. >Follows from [[Left Multiplication Linear Map of Real Matrix]]. > [!NOTE] Definition (Eigenpairs) > Consider a linear operator $\varphi:V \to V.$ We say that $\lambda\in\mathbb{R}$ is an [[Eigenpair|eigenvalue]] of $\varphi$ if there exists some nonzero $v\in V$ such that $\varphi(v)=\lambda v$We call $v$ an **eigenvector** of $\varphi$ corresponding to eigenvalue $\lambda.$ > >Let $A\in \mathbb{R}^{n\times n}$ be a [[Matrix|matrix]]. We say that $\lambda\in \mathbb{R}$ is an **eigenvalue** of $A$ if there exists some nonzero $\underline{v} \in \mathbb{R}^{n}$ such that $A\underline{v}=\lambda \underline{v}$We call $\underline{v}$ an **eigenvector** of $A$ corresponding to the eigenvalue $\lambda.$ > [!NOTE] Definition (Determinant) > For any matrix $A\in \mathbb{R}^{n\times n},$ there is an associated scalar $\det A$ called the **[[Determinant|determinant]]** of $A$ which, regarded as a function, $\begin{align} \mathbb{R}^{n\times n} &\to \mathbb{R} \\ A &\mapsto \det A \end{align}$has the following properties with respect to [[Elementary Row Operation is Equivalent to Pre-Multiplying by Elementary Matrix|elementary matrices]]: > >1. $\det(S_{ij}A)=-\det A$ >2. $\det(M_{j}(\lambda)A)=\lambda \det A$ >3. $\det(A_{ij}(\lambda)A)=\det A$ >4. $\det I_{n}=1$