Linear-1 3

Linear-1 3

Matrix addition

\begin{matrix} A \in F^{n \cdot m}; B \in F^{n \cdot m} \\ C = A \pm B \in F^{n \cdot m} \\ C_{i j} = A_{i j} \pm B_{i j} \end{matrix}

Matrix multiplication

\begin{matrix} A \in F^{m \cdot n}; B \in F^{n \cdot p} \\ A B \in F^{m \cdot p} \\ A B_{i j} = \sum_{k = 1}^{n} a_{i k} b_{k j} \end{matrix}

Properties of matrix multiplication

(A B) C = A (B C)

(A + B) C = A C + B C

C (A + B) = C A + C B

α (A B) = (α A) B = A (α B)

Special matrices

Identity matrix

I_{n} = {\begin{cases} 1 & : i = j \\ 0 & : i \neq j \end{cases}

Scalar matrix

α I = {\begin{cases} α & : i = j \\ 0 & : i \neq j \end{cases}

Diagonal matrix

D = {\begin{cases} d_{i i} & : i = j \\ 0 & : i \neq j \end{cases}

Upper triangle matrix

U = {\begin{cases} a_{i j} & : i \leq j \\ 0 & : i > j \end{cases}

Lower triangle matrix

L = {\begin{cases} a_{i j} & : i \geq j \\ 0 & : i < j \end{cases}

Exercise

\begin{matrix} A \in F^{m \cdot n} \\ Prove: A I_{n} = I_{m} A = A \end{matrix}

\begin{matrix} (A I_{n})_{i j} = \sum_{k = 1}^{n} A_{i k} I_{k j} = A_{i j} I_{j j} = A_{i j} \end{matrix}

Exercise

\begin{matrix} A, B \in F^{n \cdot n}; A, B - upper-triangle matrices \\ Prove: A B - upper-triangle matrix \end{matrix}

\begin{matrix} \forall i > j : \\ (A B)_{i j} = \sum_{k = 1}^{n} A_{i k} B_{k j} = \sum_{k = 1}^{i - 1} A_{i k} B_{k j} + \sum_{k = i}^{n} A_{i k} B_{k j} \\ \forall k < i : A_{i k} = 0 ⟹ \sum_{k = 1}^{i - 1} A_{i k} B_{k j} = 0 \\ \forall k \geq i : k \geq i > j ⟹ B_{k j} = 0 ⟹ \sum_{k = i}^{n} A_{i k} B_{k j} = 0 \\ ⟹ \forall i > j : (A B)_{i j} = 0 + 0 = 0 ⟹ A B - upper-triangle matrix \end{matrix}

Matrix trace

\begin{matrix} A \in F^{n \cdot n} \\ t r (A) = \sum_{i = 1}^{n} A_{i i} \end{matrix}

Properties of matrix trace

\begin{matrix} t r (A + B) = t r (A) + t r (B) \\ t r (α A) = α \cdot t r (A) \\ t r (A B) = t r (B A) \end{matrix}

Exercise

\begin{matrix} A \in F^{m \cdot n}; B \in F^{n \cdot m} \\ Prove: t r (A B) = t r (B A) \end{matrix}

\begin{matrix} t r (A B) = \sum_{i = 1}^{m} (A B)_{i i} = \sum_{i = 1}^{m} (\sum_{j = 1}^{n} A_{i j} B_{j i}) \\ t r (B A) = \sum_{j = 1}^{n} (B A)_{j j} = \sum_{j = 1}^{n} (\sum_{i = 1}^{m} B_{j i} A_{i j}) = \sum_{j = 1}^{n} (\sum_{i = 1}^{m} A_{i j} B_{j i}) = \\ = \sum_{i = 1}^{m} (\sum_{j = 1}^{n} A_{i j} B_{j i}) = t r (A B) \end{matrix}

Matrix transposition

\begin{matrix} A \in F^{n \cdot m} \\ A^{T} \in F^{m \cdot n} \\ (A^{T})_{i j} = A_{j i} \end{matrix}

Properties of transposition

\begin{matrix} (A^{T})^{T} = A \\ (A + B)^{T} = A^{T} + B^{T} \\ \forall A \in F^{m \cdot n} : A A^{T}, A^{T} A \in F^{k \cdot k} \\ (A B)^{T} = B^{T} A^{T} \\ \forall A \in F^{m \cdot n} : t r (A A^{T}) = 0 \leftrightarrow A = 0 \\ \forall A \in F^{n \cdot n} : t r (A^{T}) = t r (A) \end{matrix}

Exercise

Prove: \forall A \in F^{m \cdot n} : t r (A A^{T}) = 0 \leftrightarrow A = 0

\begin{matrix} 1. A = 0 ⟹ A A^{T} = 0 ⟹ t r (A A^{T}) = 0 \\ 2. t r (A A^{T}) = \sum_{i = 1}^{m} (\sum_{j = 1}^{n} A_{i j} A_{j i}^{T}) = \sum_{i = 1}^{m} (\sum_{j = 1}^{n} A_{i j}^{2}) \\ 3. t r (A A^{T}) = 0 ⟹ \forall i \in [1, m], j \in [1, n] : A_{i j}^{2} = 0 ⟹ A_{i j} = 0 ⟹ A = 0 \end{matrix}

Matrix symmetry

\begin{matrix} A \in F^{n \cdot n} \\ A = A^{T} ⟹ A is a symmetric matrix \\ A = - A^{T} ⟹ A is an anti-symmetric matrix \end{matrix}

Exercise

Prove that there are no matrices A, B \in R^{n \cdot n} such as A B - B A = I

\begin{matrix} t r (I) = n \\ t r (A B - B A) = t r (A B) - t r (B A) = t r (A B) - t r (A B) = 0 \\ I = A B - B A ⟹ t r (I) = t r (A B - B A) ⟹ n = 0 \\ ⟹ There are no such matrices A, B \end{matrix}