Matrices

• An $m\times n$ matrix with entries from F is a rectangular array
  • Notated $columns\times rows$
• Rows are vectors in $F^n$, columns in $F^m$
• Zero matrix has zeros in each entry
• $m\times n$ matrices are equal if $A_{ij}=B_{ij}$

Vector Spaces

• The set of all $m\times n$ matrices with entries in F is a vector space denoted $M_{m\times n}(F)$
  • $A,B\in M_{m\times n}(F),c\in F$
  • $(A+B{)}_{ij}=A_{ij}+B_{ij}$
  • $(cA{)}_{ij}=c{A}_{ij}$

Operations

Matrix Multiplication Elementary Matrix Operations

Transposition

• Transpose $A^t$ is the matrix with elements $(A^t{)}_{ij}=A_{ji}$
• Flipped over the diagonal
• Properties
  • $(A+B{)}^t=A^t+B^t$
  • $(AB{)}^t=B^t{A}^t$
• Conjugate Transpose (or adjoint)
  • $A^{\ast }$ such that $(A^{\ast }{)}_{ij}=\overline{A_{ji}}$

Augmentation

• Let $A\in M_{m\times n}(F),B\in M_{m\times p}$. The augmented matrix is the $m\times (n+p)$ matrix $(A|B)$ that you get if you just stick them together
• $M,A,B\in M_{n\times n}(F).M(A|B)=(MA|MB)$

Categories

• A symmetric matrix $A$ is such that $A^t=A$
• A $n\times n$ matrix $A$ is diagonal if $A_{ij}=0$ whenever $i\ne j$

Rank

• Nullspace and Range
• $\text{rank}(A),A\in M_{m\times n}(F)$ is the rank of $L_A:F^n\to F^m$
• $\text{rank}(T)=\text{rank}([T{]}_{\beta }^{\gamma })$ where $T:V\to W$ linear and finite dimensional
• Properties
  • Rank is not affected by a multiplication by an invertible matrix
  • The rank of the matrix equals the number of linearly independent columns
    • $\text{rank}(A)=\dim (\text{span}(a_1,\dots ,a_n))$
  • $\text{rank}(A^t)=\text{rank}(A)$
    • → rank is also the number of linearly independent rows
  • $\text{rank}(AB)\le \text{rank}(A)$ and $\text{rank}(AB)\le \text{rank}(B)$
  • $A\in M_{m\times n},\text{rank}(A)=r$. Then $r\le m,r\le n$ and a finite number of elementary row and column operations can transform a into the diagonal matrix $D=\left(\begin{array}{cc}{I}_r& 0_1\\ 0_2& 0_3\end{array}\right)$ and there exists invertible $B,C$ such that $D=BAC$

Extra

Kronecker Delta

$${\delta }_{ij}=\{\begin{array}{l}1\text{ if }i=j\\ 0\text{ if }i\ne j\end{array}$$