# Types of matrices

Defining the basic types of matrices is essential to be able to build other much more complex types and methods.

The base is essential. And when we speak of base we are not referring to any mathematical concept. We are referring to the knowledge base. Matrices are one of the most important and widely used concepts in different fields of science.

In econometrics, in computer programming, in big data and in various fields in which it is a question of crossing data or working with a large amount of data.

## Square matrix

A square matrix satisfies that (m = n). In other words, it has the same number of rows and columns. So the dimension of the rows will be the same as the dimension of the columns.

The square matrix is very important because it is the basis for many matrix types and methods.

Example

Matrix dimension B = 2 x 2.

## Transposed Matrix

A transposed matrix consists of reordering the original matrix by changing the rows by columns and the columns by rows.

Generally, a transposed matrix is indicated with a superscript T or an apostrophe ('). To express it better, we opted for the superscript T.

Following the previous example it would be: BT.

Example

When the original matrix is a square matrix, as in our case, the dimension of the matrix remains the same because the number of rows and columns is the same.

Matrix dimension BT = 2 x2.

## Identity Matrix

The identity matrix is a square matrix in which all its elements are zeros except those that belong to its main diagonal. It is usually identified by the letter I.

The identity matrix can be quickly distinguished without doing any calculations.

We have assigned a 3 × 3 dimension in this case. However, this dimension can be larger or smaller. We only have to comply when the matrix remains square and meets the characteristic: all zeros except its main diagonal which must have ones.

Example

The identity matrix acts like the number 1 in common algebra. Let I be the identity matrix and B any matrix, the product of both has a neutral effect on matrix B. Therefore, matrix B is the same as IB.

## Triangular Matrix

A triangular matrix is a square matrix in which the elements below the main diagonal are zeros or the elements above the main diagonal are zeros.

The triangular matrix focuses on the location of *triangles* containing only zeros. Depending on its position with respect to the main diagonal, the triangular matrix will be called upper or lower.

Upper triangular matrix:

Lower triangular matrix (lower):

The triangular matrix participates in the Lower-Upper (LU) decomposition method, which is used to obtain the Cholesky decomposition. This method is widely used in quantitative finance to transform independent normal variables into correlated normal variables.

## Symmetric Matrix

A matrix is symmetric if it is a square matrix and coincides with its transpose (C = CT).

To find symmetric matrices in a simple way, we just have to look at the element triangles that are above and below the main diagonal.

Example

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