Matrix

Matrix

Definition 14.
(a) An $m \times n$ matrix $A$ with entries from a field $F$ is a rectangular array of the form
$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$ where each entry $a_{ij}\, (1 \leq i \leq m, 1 \leq j \leq n)$ is an element of $F$. The number $a_{ij}$ is called the $(i, j)$-entry of the matrix $A$, and is also written as $a_{ij} = [A]_{ij}$.
(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 $A$ is called square. In contrast, if $m \neq n$, then the matrix $A$ is called rectangular. Moreover, the matrix $A$ is said to be wide if $m \leq n$, and is long if $m \geq n$.
(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 $j$.
(h) The $n \times n$ identity matrix $I_n$ (or $I$ if the size is clear from the context) is a diagonal matrix whose diagonal entries are all $1$. 

Matrix space

Theorem 19. 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 $1 \leq i \leq m, 1 \leq j \leq n$.

Definition 15

Definition 15. 
(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 $i > j$, and a lower triangular matrix is an $m \times n$ matrix such that $A_{ij = 0}$ whenver $i < j$.

Diagonal matrix와 trace는 오로지 square, 즉 정사각행렬에 대해서만 정의된다.

Theorem 20

Theorem 20. 
(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 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 $M_{n \times n}(F)$.

Matrix Multiplication

Definition 16. Let $A \in M_{m \times n}(F), B \in M_{n \times p}(F)$. We define the product of $A$ and $B$, denoted $AB$, to be the $n \times p$ matrix such that $$(AB)_{ij} = \sum_{k=1}^{n}A_{ik}B_{kj} \,\, \text{for} \,\, 1 \leq i \leq m, 1 \leq j \leq p.$$

    행렬의 곱셈은 앞서 합과 상수배를 정의했던 것처럼 성분별로 정의되지 않고 언뜻 보면 부자연스럽게 정의된다. 때문에 일반적으로 교환법칙도 성립하지 않는데, 이는 행렬곱은 사실 선형변환의 합성과 대응되도록 정의되었기 때문이다.

Theorem 21

Theorem 21. Let $A \in M_{m \times n}(F), B, C \in M_{n \times p}(F)$, and $D, E \in M_{q \times m}(F)$. Then
(a) $A(B + C) = AB + AC$ and $(D + E)A = DA + EA.$
(b) $a(AB) = (aA)B = A(aB), \, \forall a \in F.$
(c) $I_mA = A = AI_n.$
(d) Let $X, Y$, and $Z$ be matrices such that $X(YZ)$ is defined. Then $(XY)Z$ is also defined and $X(YZ) = (XY)Z$.

Proof. (d) To define $X(YZ)$, suppose that $X, Y$, and $Z$ are an $m \times n$, $n \times p$, $p \times q$ matrices, respectively. Then $X(YZ)$ is an $m \times q$ matrix, and so is $(XY)Z$. 
We have $L_{X(YZ)} = L_X(L_YL_Z) = (L_XL_Y)L_Z = L_{(XY)Z} \Longrightarrow X(YZ) = (XY)Z.$ $\blacksquare$

One-Sided Inverse of a Matrix

Definition 17. Let $A \in M_{m \times n}(F)$. Then $B \in M_{n \times m}(F)$ is called a left inverse of $A$ if $BA = I_n$ and $C \in M_{n \times m}(F)$ is called a right inverse of $A$ if $AC = I_m$. 

Remark

Remark. For $M_{m \times n}(F)$, the followings hold:
(a) The left and right identities are not the same in general.
(b) There may be infinitely many one-sided inverses.
(c) 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 $B = C$.

Inverse of a Matrix

Definition 18. Let $A \in M_{n \times n}(F)$. Then $A$ is invertible (or nonsingular) if $\exists B \in M_{n \times n}(F)$ such that $AB = BA = I_n$. The matrix $B$ is called the inverse of $A$ and is denoted by $A^{-1}$.

Exponent of a Matrix

Definition 19. Let $A \in M_{n \times n}(F)$. Definte $A^{0} = I_n$. For any positive integer $k$, $A^k$ is inductively defined as $A^k = A(A^{k-1})$. If $A$ is invertible, then $A^{-k}$ is also defined as $A^{-k} = (A^{-1})^k$.

Remark. It is easy to check that $A^{k+l} = A^kA^l$ whenever the RHS is defined. (Note that if $A$ is not invertible, then $A^{3 + (-1)}$ is defined but $A^3A^{-1}$ is not.)

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

Linear Algebra | Stephen Friedberg - 교보문고