matrix

 Types of Matrix

Row Matrix

A row matrix is formed by a single row.

Column Matrix

A column matrix is formed by a single column.

Rectangular Matrix

A rectangular matrix is formed by a different number of rows and columns, and its dimension is noted as: mxn.

Square Matrix

A square matrix is formed by the same number of rows and columns.

The elements of the form a_ii constitute the principal diagonal.

The secondary diagonal is formed by the elements with i+j = n+1.

Zero Matrix

In a zero matrix, all the elements are zeros.

Upper Triangular Matrix

In an upper triangular matrix, the elements located below the diagonal are zeros.

Lower Triangular Matrix

In a lower triangular matrix, the elements above the diagonal are zeros.

Diagonal Matrix

In a diagonal matrix, all the elements above and below the diagonal are zeros.

Scalar Matrix

A scalar matrix is a diagonal matrix in which the diagonal elements are equal.

Identity Matrix

An identity matrix is a diagonal matrix in which the diagonal elements are equal to 1.

Transpose Matrix

Given matrix A, the transpose of matrix A is another matrix where the elements in the columns and rows have switched. In other words, the rows become the columns and the columns become the rows.

(A^t)^t = A

(A + B)^t = A^t + B^t

(α ·A)^t = α · A^t

(A · B)^t = B^t · A^t

Regular Matrix

A regular matrix is a square matrix that has an inverse.

Singular Matrix

A singular matrix is a square matrix that has no inverse.

Idempotent Matrix

The matrix A is idempotent if:

A² = A.

Involutive Matrix

The matrix A is involutive if:

A² = I.

Symmetric Matrix

A symmetric matrix is a square matrix that verifies:

A = A^t.

Antisymmetric Matrix

An antisymmetric matrix is a square matrix that verifies:

A = −A^t.

Orthogonal Matrix

A matrix is orthogonal if it verifies that:

A · A^t = I.