Matrix Operations

Transposition §

• mirror matrix across main diagonal (upper left to bottom right)
• ${\mathbf{A}}_{i,j}^{\top }={\mathbf{A}}_{j,i}$
• For vectors, columns -> rows, rows -> columns

Addition, Multiplication §

• works with same shape, just add element-wise

Broadcasting §

• vector is filled to form a matrix of same size
• implicit when adding vector to matrix of same dimension on one side

Multiplying Matrices and Vectors §

• matrix product is just dot product but with each vector
• element-wise product is different
• distributive
• associative
• NOT commutative for matrices, yes for vectorsA
• $(AB{)}^{\top }=B^{\top }{A}^{\top }$
• system of linear equations: $\mathbf{A}x=b$, where $x=[x_1,x_2,\dots ,x_n]$ is a set of variables we are solving for

$$\begin{array}{r}3x+2y+4z=-1\\ 4y+z=3\\ 2x+6y+2z=4\end{array}$$

becomes

$$\begin{array}{c}\left[\begin{array}{ccc}3& 2& 4\\ 0& 4& 1\\ 2& 6& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}-1\\ 3\\ 4\end{array}\right]\end{array}$$