Matrix

Vectors in form of matrix:

$$\left(\begin{array}{c}\left(\begin{array}{c}1\\ 3\\ 5\end{array}\right),\left(\begin{array}{c}2\\ 4\\ 6\end{array}\right)\end{array}\right)\to \left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$$

NOTE

A $m\times n$ matrix is a table with $m$ rows and $n$ columns also noted as $A=[a_{ij}{]}_{i=1,j=1}^{mn}\in {\mathbb{R}}^{m\times n}$ OR we can treat a matrix as a function: $A:[m]\times [n]\to \mathbb{R}$ $A(i,j)=A_{ij}$

other notation form $A=\left[\begin{array}{c}{\mathbf{v}}_1,\dots {\mathbf{v}}_n\end{array}\right]$ $A=\left[\begin{array}{c}{\mathbf{u}}_1^{\top }\\ ⋮\\ {\mathbf{u}}_m^{\top }\end{array}\right]$ $0$ : The zero matrix

Identische matrix $I$ $a_{ij}={\delta }_{ij}$ with Kronecker-Delta ${\delta }_{ij}=1$ falls $i=j$ und $0$ sonst

NOTE

Die Spalten von $A$ sind linear unabhängig $x=0$ ist der einzige Vektor mit $Ax=0$ Other equivalent statements

Different views of matrix multiplication

1. Column-vector (transform) view: A acts on each column of B

   • Write $B=[b_1{b}_2\dots b_p]$ with $b_j\in {\mathbb{R}}^n$
   • Then $C=AB=[A{b}_{1}A{b}_2\dots A{b}_p]$
   • Interpretation: same linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ applied to every column of $B$

2. Column-combination view: each column of C is a linear combination of columns of A

   • Write $A=[a_1{a}_2\dots a_n]$ with $a_k\in {\mathbb{R}}^m$
   • For each column $j$: $c_j=A{b}_j={\sum }_{k=1}^n{b}_{kj}{a}_k$
   • Interpretation: coefficients come from the entries of column $b_j$ of $B$

3. Row-vector (transform) view: each row of C is a row of A acting on B

   • Write $A=\left[\begin{array}{c}{\alpha }_1^T\\ {\alpha }_2^T\\ ⋮\\ {\alpha }_m^T\end{array}\right]$ with ${\alpha }_i^T\in {\mathbb{R}}^{1\times n}$
   • Then $C=\left[\begin{array}{c}{\alpha }_1^{T}B\\ {\alpha }_2^{T}B\\ ⋮\\ {\alpha }_m^{T}B\end{array}\right]$
   • Interpretation: “row times matrix” = precompose a measurement with $B$; also $({\alpha }_i^{T}B{)}^T=B^T{\alpha }_i$

4. Row-combination view: each row of C is a linear combination of rows of B

   • Write $B=\left[\begin{array}{c}{r}_1\\ r_2\\ ⋮\\ r_n\end{array}\right]$ with $r_k\in {\mathbb{R}}^{1\times p}$
   • For each row $i$: $c_i^T={\alpha }_i^{T}B={\sum }_{k=1}^n{a}_{ik}{r}_k$
   • Interpretation: coefficients come from the entries of row ${\alpha }_i^T$ (row $i$ of $A$)

Column space

Definition

$\mathbf{C}(A):=\{Ax:x\in {\mathbb{R}}^n\}\subseteq {\mathbb{R}}^m$

$$\mathbf{C}(\left[\begin{array}{cc}2& 3\\ 3& -1\end{array}\right])=Span\left(\left(\begin{array}{c}2\\ 3\end{array}\right),\left(\begin{array}{c}3\\ -1\end{array}\right)\right)={\mathbb{R}}^2$$

![[Linear Independence#Span of vectors#Definition of Span 1.25]]

Rank

$rank(A)=n$ : linear unabhängige Spalten

if $rank(A)=0$ : Nullmatrix

Example $rank(\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right])=1$

Lemma 2.11

Transpose (Transposition)

$A^{\top }=[a_{ji}{]}_{i=1,j=1}^{nm}$

Example

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]\leftrightarrow A^{\top }=\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$$ $$(A^{\top }{)}^{\top }=A$$ $$(A+B{)}^{\top }=A^{\top }+B^{\top }$$ $$(AB{)}^{\top }=B^{\top }{A}^{\top }$$

Row Space

$\mathbf{R}(A):=\mathbf{C}({\mathbf{A}}^{\top })$

Note that

• $\mathbf{R}(A)\subseteq {\mathbb{R}}^n$
• $\mathbf{C}(\mathbf{A})\subseteq {\mathbb{R}}^m$

Null Space

$0\in N(A)$

$N(A)=\{0\}⟺$ Spalten von A sind linear unabhängig $N(A)={\mathbb{R}}^n⟺A=0$

Important

If the columns of a matrix are linearly independent, then the null space contains only the zero vector. consider $A\mathbf{x}=x_1{\mathbf{a}}_1+x_2{\mathbf{a}}_2+\cdots +x_n{\mathbf{a}}_n$

Properties

Lemma 5.1.10

Title

$N(A)=N(A^{\top }A)$ and $C(A^{\top })=C(A^{\top }A)$

Proof: $N(A)\subseteq N(A^{\top }A)$ $x\in N(A)$ then $Ax=0$ $⟹A^{\top }Ax=A^{\top }0=0$ $⟹x\in N(A^{\top }A)$

$N(A^{\top }A)\subseteq N(A)$ $x\in N(A^{\top }A)$ then $A^{\top }Ax=0$ $x^{\top }{A}^{\top }Ax=(Ax{)}^{\top }Ax=\Vert Ax{\Vert }^2=0⟹Ax=0⟹x\in N(A)$

$⟹N(A)=N(A^{\top }A)$

Link to original

Trace

The trace of a matrix is the sum of its diagonal elements.

$$\text{tr}(A)=\sum _{i=1}^n{a}_{ii}$$

Let $A$ be a square matrix and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be its eigenvalues, then: $\mathrm{Tr}(A)={\sum }_{i=1}^n{\lambda }_i$ $\det (A)={\prod }_{i=1}^n{\lambda }_i$