Vector

Each vector is an object with many features values.

Matrix

Rows are an object with many features, Alice’s age gender height weight. Columns are Features for many objects, age of Alice Bob Chris David.

$$ A = \begin{bmatrix} \vec{c}_1 & \vec{c}_2 & \cdots & \vec{c}_n \end{bmatrix} $$

$\vec{c}_1, \vec{c}_2, \cdots, \vec{c}_n$ are the base vectors after transformation described in current axis.

Apply matrix to a vector $\vec{v}$ means you get what $\vec{v}$ means in current axis.

So $A$ is information in current, $\vec{v}$ is information in transformed, $A\vec{v}$ is information in current.

So $\vec{v}$ is hola and $A\vec{v}$ is hello.

$$ A = U\Sigma V^{T} $$

So any matrix is: Rotate → Scale → Rotate

It means that we transform information to something we are familiar with, to somewhere many fluent tools are available, then goes back to another encoding system.

Determinate

$$ \det(A) = \prod_{i=1}^{n} \lambda_i $$

determinant tells you how much a matrix scales space

if $\det(A)=0$, the transformation will squish all of the space into a lower dimension. This means some information is destroyed in the process. And some of the axis are redundant information.

All columns are linearly independent.

Inverse

$$\mathrm{rank}(A) = n$$

Rank

Non-square

$$A \in \mathbb{R}^{m \times n}, \quad B \in \mathbb{R}^{p \times q}$$

the number of columns in $A$ must equal the number of rows in $B$, that is.

$$ n = p $$

and then

$$ AB \in \mathbb{R}^{m \times q} $$

Eigenvector

$$ A\vec{v} = \lambda \vec{v} $$

A pure rotation matrix doesn’t have eigenvectors.

$$ \text{det}(A -\lambda I) = 0 $$

so it squish to a lower dimension.