Matrix Algebra

consisting of rows and columns of “elements”. Also more dimensions are possible.

$M=(r\times c)$

Square Matrix … same amount of rows and columns $r=c$ Vector … a matrix with one column $(r\times 1)$ Column Vector … a matrix with one row $(1\times c)$

$a_{ij}$ → i … row, j … column

Why matrices are useful?

Special Matrices

convexity and concavity of multi-function variables

• Jacobian Matrix
• Hessian Matrix

Identity Matrix

1 0 0
0 1 0
0 0 1

Calculating with Matrices

Equality

• same size (rows and columns)
• all elements are the same (position and value)

Addition

• same size
• add both elements at the respective position
• Assoziativ → $(A+B)+C=A+(B+C)$
• Kommunikativ → $A+B=B+A$
• Distributiv → $(\alpha +\beta )\ast A=\alpha A+\beta A$

Multiplication

• first matrix columns = second matrix rows
  • $A_{3\times 2},B_{2\times 3}$
  • inner pair → possible? AB = true, BA = true
  • outer pair → size … $AB=3\times 3$, $BA=2\times 2$
• wontfix get visualiazation from 3b1b
• Assoziativ
• Kommunikativ
• Distributiv
• only a square matrix can be indefinetely raised to a power
• $(A+B{)}^2=A^2+AB+BA+B^2\ne A^2+2AB+B^2$ … $BA\ne AB$

Example Market Share

wontfix copy text from slides

$T=(...)$

• columns: how much each company looses
• rows: how much each company receives #wontfix get resulting matrix from phone

Example Airports

• A to B
  • 2 rows … from A
  • 4 columns … to B
• B to C
  • 4 columns … from B
  • 3 rows … to C
• resulting matrix $A\to C=AB\ast BC...2\times 3$

Transposition

• “mirror” the matrix along it’s diagonal
• swapping the rows and columns
• $a_{ij}^{\prime }=a_{ji}$
• $(A^{\prime }{)}^{\prime }=A$
• Distributive with Sum
• Distributive with swapped Mulitplication
  • $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$
• symmetric … $A^{\prime }=A$
  • square matrix
  • Proof that $X^{\prime }X$ and $X{X}^{\prime }$ are symmetric
    • $(X^{\prime }X{)}^{\prime }=X^{\prime }(X^{\prime }{)}^{\prime }=X^{\prime }X$

Gaussian Elimination

For solving systems of equations.

Example Fish & Lumber

In the normal way

• $F:x_1=d_1+\beta x_2$
• $T:x_2=d_2+\delta x_3$
• $B:x_3=\alpha x_1$
• after substituting twice in $x_1$ and using values we get
• $x_1=100+20+\frac{1}{4}{x}_1$
  • $x_1=160$
  • $x_2=240$
  • $x_3=80$

Now with a Matrix

• transpose the equations to have variables on the left and constants on the right
• collect all coefficients from the system of equations

x_1   x_2  x_3 | result
---------------|--------
 1   -.25    0 |    100
 0      1   -2 |     80
-1      0    2 |      0

• then start to eliminate to lead to the left side being an identity matrix.

• you can find another explanation and visualization here: https://www.youtube.com/watch?v=2tlwSqblrvU (5 mins)

Equivalence

Equivalence between such systems in matrix notation are defined with a Tilde: ~

Inverse Matrix

Inverse of scalar numbers … $\frac{1}{\alpha }\ast \alpha =1$

The same is true with the inverse matrix … $X$.

$AX=XA=1$

$X...A^{-1}$

If an inverse matrix exists, then the matrix is “invertible”, if there is no inverse matrix it is not invertible.

• $(A^{-1}{)}^{-1}=A$
• $(AB{)}^{-1}=B^{-1}{A}^{-1}$
• $(A^{\prime }{)}^{-1}=(A^{-1}{)}^{\prime }$
• $(\alpha A^{-1})=\frac{1}{\alpha }\ast A^{-1}$

Solve

a b | 1 0
c d | 0 1