Week3-4 - Matrix Algebra

Here we note A a matrix nxm,
n: number of rows
m: number of columns

In R, to create a matrix:

matA = matrix(data=c(1,2,3,4,5,6),nrow=2,ncol=3)

To create a vector:
```
xvec = c(1,2,3)
```
To convert:
```
xvec.mat = as.matrix(xvec)
```
To transpose a matrix or vector:
```
t(mat)
t(xvec)
```
A matrix A is symetric if \(A=A^T\).

Matrix multiplication is not commutative: \(A \cdot B \neq B \cdot A\)
Matrix multiplication in R:
```
 A %*% B
```
nxn identity matrix in R is created with:
```
diag(n)
```
In R to inverse a matrix we use:
```
invA = solve(A)
```
For a 2×2 matrix we can compute the inverse as:

\[A^{-1} = \frac{1}{det(A)} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix}, ~~ \text{assuming: } det(A) = a_{11}a_{22} - a_{12}a_{21} \neq 0\]

If we have a vector of 1 called one, then we can use: \(one^T x = \sum\limits_{i=1}^n x_i\)
In R we have the function crossprod(x,y) = \(x^Ty\)

We consider 3 assets with return rates \(R_i\) normally distributed.
The portfolio is normally distributed too. We can compute its mean and variance already.
We then note: \(R = \begin{bmatrix} R_A \\ R_B \\ R_C \end{bmatrix}\), \(\mu = \begin{bmatrix} \mu_A \\ \mu_B \\ \mu_C \end{bmatrix}\), \(\mathbf{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\), \(x = \begin{bmatrix} x_A \\ x_B \\ x_C \end{bmatrix} \)
And finally \(\Sigma = \begin{bmatrix} \sigma_A^2 & \sigma_{AB} & \sigma_{AC} \\ \sigma_{AB} & \sigma_B^2 & \sigma_{BC} \\ \sigma_{AC} & \sigma_{BC} & \sigma_C^2 \end{bmatrix} \)

Portfolio weights sum to 1 so we have : \(\mathbf{1}^Tx = 1\)

\(R_{p,x} = x^T R = R^T x\)

\(\mu_{p,x} = x^T\mu = \mu^T x\)

\(\sigma_{p,x}^2 = x^T \Sigma x\)

2 portfolio with weights x and y,
The covariance between the 2 is: \(Cov(R_{p,x},R_{p,y}) = x^T \Sigma y = y^T \Sigma x\)

We note: \(\mathbf{X} = \begin{bmatrix} X \\ Y \end{bmatrix}, ~ x = \begin{bmatrix} x \\ y \end{bmatrix}, ~ \mu = \begin{bmatrix} \mu_X \\ \mu_Y \end{bmatrix}, \Sigma = \begin{bmatrix} \sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} & \sigma_Y^2 \end{bmatrix}\)