Matrix

A matrix is an array of numbers. This below is a 3x2 matrix.

$$A= \left[ \begin{matrix} a_1 & a_2 \\ a_3 & a_4 \\ a_5 & a_6 \end{matrix} \right]$$

notes

A vector is a n*1 matrix.

Adding

$$\left[ \begin{matrix} a_1 & a_2 \\ a_3 & a_4 \\ a_5 & a_6 \end{matrix} \right] + \left[ \begin{matrix} b_1 & b_2 \\ b_3 & b_4 \\ b_5 & b_6 \end{matrix} \right] = \left[ \begin{matrix} a_1+b_1 & a_2+b_2 \\ a_3+b_3 & a_4+b_4 \\ a_5+b_5 & a_6+b_6 \end{matrix} \right]$$

::: notes note

The two matrices must be the same size

:::

Multiplying

To multiplying a single number

To multiplying another matrix (dot product)

Identity Matrix

Identity matrix is always "Square". It has 1s on the main diagonal and 0s everywhere else. Its symbol is the capital letter $I$.

Example: 3x3 identity matrix.

$$I=\left[ \begin{matrix} 1 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]$$

For matrix $A$ and Identity matrix $I$.

$$A * I = A \\ I * A = A$$

Note

For matrix $A$ and $B$.

$$AB \neq BA$$

Inverse of a Matrix

Just like a number has a reciprocal.

$$8 \rightarrow \frac{1}{8} \\ A \rightarrow A^{-1}$$

We write $A^{-1}$ instead of $\frac{1}{A}$ because we don't divide by a matrix! The inverse of $A$ is $A^{-1}$ only when:

$$AA^{-1}=I=A^{-1}A$$

Orthogonal Matrix

Orthogonal Matrix is a real square matrix (opens new window) whose columns and rows are orthonormal (opens new window) vectors (opens new window). One way to express this is:

$$AA^{\top}=I=A^{\top}A$$

Transposing

To "transpose" a matrix, swap the rows and columns.

We put a "T" in the top right-hand corner to mean transpose:

$$\left[ \begin{matrix} a_1 & a_2 \\ a_3 & a_4 \\ a_5 & a_6 \end{matrix} \right] ^\top = \left[ \begin{matrix} a_1 & a_3 & a_5 \\ a_2 & a_4 & a_6 \end{matrix} \right]$$

References

Matrix Multiplying (opens new window)

Matrix (opens new window)

Inverse of a Matrix (opens new window)

深度学习和机器学习的线性代数入门 | 雷锋网 (leiphone.com) (opens new window)