What is it?
Matrices are used in Mathematics, mainly in Linear Algebra to represent a mathematical object, or the property of said subject. They can store various different values inside of them.
Matrices can be thought of just like tables. They have a two-dimensional projection with elements organized in rows and columns. It forms a rectangular array, which in itself, can also be used to build matrices.****
A special case of a matrix: Vectors
Vectors are a special case of a matrix where a matrix has only a single column. It may not seem to special, but it’s probably the most used matrix in Mathematics.
A common notation is , where is the matrix variable, the number of rows and the number of columns. We can also express a filled matrix like this:
You can also identify each element of the matrix, using the notation , where is the element name. Applied to the above matrix:
A_{2\times3}= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} $$___ # Special cases of matrices *Matrices* can be given **different names** if they are presented in a given way. These are some of those special cases: ## Diagonal matrix A *square matrix*, with **equal length in both sides**, which the **elements that don't pertain to the main diagonal** are equal to $0$, is called a *diagonal matrix*. For example:A_{2\times2} = \begin{bmatrix} -10 & 0 \ 0 & 4 \end{bmatrix} \qquad \qquad B_{3\times3} = \begin{bmatrix} 3 & 0 & 0 \ 0 & 12 & 0 \ 0 & 0 & 2 \end{bmatrix}
## Identity matrix An *identity matrix* is similar to the *diagonal matrix*, but the elements from the **main diagonal are equal to $1$**, and the rest must be **equal to $0$**.I_{2} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \qquad \qquad I_{3} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}
___ ## Transposed matrix To **transpose a *matrix***, change its *rows* by the *columns*, and the *columns* in the place of the *rows*, represented by an upper $T$.A = \begin{bmatrix} 20 & 16 & 5 \ 2 & 6 & 11 \end{bmatrix} \qquad \qquad A^T = \begin{bmatrix} 20 & 2 \ 16 & 6 \ 5 & 11 \end{bmatrix}
___ ## Triangular matrices If the elements **below the main diagonal** are all equal to $0$, we have an *upper triangular matrix*. If the elements **above the main diagonal** are all equal to $0$, we have a *lower triangular matrix*. For example:\text{Upper triangular} = \begin{bmatrix} 8 & 33 & 11 \ 0 & 6 & 12 \ 0 & 0 & 21 \end{bmatrix} \qquad \text{Lower triangular} = \begin{bmatrix} 5 & 0 \ 7 & 1 \end{bmatrix}
___ # Matrix Operations Some basic operations like **sum and subtraction** are made on an element-by-element basis in *matrices*. However, other operations like [[Matrix Multiplication]] have their specific methods.