Matrix

Definition 1

Definition 1.
(a) An $m \times n$ matrix $A$ with entries from a field $F$ is a rectangular array of the form
$$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$$ where each entry $a_{ij}\, (1 \leq i \leq m, 1 \leq j \leq n)$ is an element of $F$. We often denote $A = [a_{ij}]$ or $a_{ij} = [A]_{ij}$. 
(b) The entries $a_{ij}$ with $i=j$ are the diagonal entries of the matrix.
(c) The entries $a_{i1}, a_{i2}, \cdots, a_{in}$ compose the ith row of the matrix, and the entries $a_{1j}, a_{2j}, \cdots, a_{mj}$ compose the jth column of the matrix. The $m \times n$ matrix in which each entry equals zero is called the zero matrix, denoted by $O$.
(d) If $m = n$, that is, the number of rows and columns of a matrix are equal, then the matrix is called square.
(e) Two matrices $A$ and $B$ are said to be equal if they have the same sizes and $A_{ij} = B_{ij}$ for all $i$ and $j$.

Matrix space

Theorem 1. The set of all $m \times n$ matrices with entries from a field $F$ is a vector space, denoted $M_{m \times n}(F)$, with the following operations of matrix addition and scalar multiplication: For $A, B \in M_{m \times n}(F)$ and $c \in F$,
$$\begin{align*} (A+B)_{ij} = A_{ij}+B_{ij} \,\,\text{and}\,\, (cA)_{ij}=cA_{ij} \end{align*}$$ for $1 \leq i \leq m, 1 \leq j \leq n$.
We denote the set of all $n \times n$ square matrices with entries from $F$ by $M_{n}(F)$ instead of $M_{n \times n}(F)$.

Remark

Remark. For $M_{m \times n}(F)$, the followings hold:
(1) The left and right identities are not the same in general. 
(2) There may be infinitely many one-sided inverses. 
(3) If $m = n$, the left and right inverses are equal. That is, for $A \in M_{n}(F),$if $\exists B, C \in M_{n}(F)$ such that $BA = I_n$ and $AC = I_n$, then $B = C$. 

Definition 2

Definition 2. 
(a) The transpose $A^T$ of an $m \times n$ matrix $A$ is the $n \times m$ matrix obtained from $A$ by interchanging the rows with the columns, i.e., $(A^T)_{ij} = A_{ji}$.
(b) A symmetric matrix is a matrix $A$ such that $A^T = A$. 
(c) A skew-symmetric matrix is a matrix $A$ if $A^T = -A$.
(d) A diagonal matrix is an $n \times n$ matrix $M$ such that $M_{ij} = 0$ whenever $i \neq j$.
(e) The trace of an $n \times n$ matrix $M$, denoted by tr($M)$, is the sum of the diagonal entries of $M$, that is,
$$\text{tr}(M) = M_{11} +M_{22} + \cdots + M_{nn}.$$ (f) A upper triangular matrix is an $m \times n$ matrix such that $A_{ij} = 0$ whenever $i > j$.

    diagonal matrix와 trace는 오로지 square, 즉 정사각행렬에 대해서만 정의된다.

Theorem 2

Theorem 2. 
(a) $(aA+bB)^T = aA^T + bB^T$ for any $A, B \in M_{m \times n}(F)$ and any $a, b\in F$. 
(b) $(A^T)^T = A$ for each $A \in M_{m \times n}(F)$.
(c) $A + A^T$ is symmetric for any square matrix $A$.
(d) $A - A^T$ is skew-symmetric for any square matrix $A$.
(e) tr($aA+bB$) = $a$ tr($A$) + $b$ tr($B)$ for any $A, B \in M_{n \times n}(F)$.

Remark

Remark. 
(a) Every square matrix can be uniquely expressed by a sum of a symmetric matrix and skew-symmetric matrix. 
$(\because) A = \frac{A + A^T}{2} + \frac{A - A^T}{2}.$
(b) A matrix $A$ is skew-symmetric $\iff$ $a_ii = 0$ for all $i$ and $a_{ij} = -a_{ji}$ for all $i$ and $j$ such that $i \neq j$.
(c) The set $W$ of all symmetric matrices in $M_{n \times n}(F)$ is a subspace of $M_{n \times n}(F)$. 
(d) The set of diagonal matrices is a subspace of $M_{n \times n}(F)$.
(e) The set of $n \times n$ matrices having trace equal to zero is a subspace of $M_{n \times n}(F)$.

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Reference is here: https://product.kyobobook.co.kr/detail/S000003155051

Linear Algebra | Stephen Friedberg - 교보문고