# Matrix Math Basic Concept

**Matrix:**

A matrix is a rectangular arrangement of numbers, expressions, or symbols in rows and columns enclosed by ( ), [ ], or || ||.

**Row:** The row elements are horizontally arranged

**Column:** The column elements are vertically arranged.

What is the **Dimension** of a Matrix?

The dimensions, or size, of a matrix, are defined as, [Number of Rows * Number of Columns ]

A matrix is usually denoted by a capital letter and the elements within the matrix are denoted by lowercase letters,

**Row Matrix:**

A matrix is said to be a row matrix if it has only one row. A row matrix has only one row but any number of columns.

**Column Matrix:**

A matrix is said to be a column matrix if it has only one column. A column matrix has only one column but any number of rows.

**Square Matrix:**

A square matrix has the number of columns equal to the number of rows.

**Rectangular Matrix:**

A matrix where number of rows is not equal to the number of columns is called rectangular matrix.

**Diagonal Matrix:**

A square matrix is said to be a diagonal matrix when – 1. aij != 0 (i=j) | 2. aij = 0 (i!=j)

A square matrix where all the elements are zero except those on the main diagonal.

**Scalar Matrix:**

A square matrix is said to be a scalar matrix when – 1. aij = k (i=j) | 2. aij = 0 (i!=j)

A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.

**Identity/Unit Matrix:**

A square matrix is said to be a scalar matrix when – 1. aij = 1 (i=j) | 2. aij = 0 (i!=j)

An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else and denoted by I.

**Triangular Matrix:**

A square matrix is said to be a triangular matrix when – 1. aij = 0 (i<j) | 2. aij = 0 (i>j)

A square matrix whose elements above or below the main diagonal are all zero

**Upper Triangular Matrix:**

A square matrix is said to be a upper triangular matrix when – 1. aij = 0 (i>j)

A square matrix whose elements below the main diagonal are all zero.

**Lower Triangular Matrix:**

A square matrix is said to be a upper triangular matrix when – 1. aij = 0 (i<j)

A square matrix whose elements above the main diagonal are all zero.

**PERMUTATION MATRIX**

A square matrix is said to be a permutation matrix when –

1. Have to be a square matrix

2. Each entry either 1 or 0

3. Each row contain a significance one (1)

4. Each column contain a significance one (1)

**Sparse Matrix:**

A square matrix is said to be sparse matrix when – 1. Non-zero > Zero

**Dense matrix:**

A square matrix is said to be dense matrix when – 1. Zero > Non-zero

**Scalar Multiplication:** It is defined with, “multiply each element in the matrix by the given scalar”

**Matrix Addition & Subtraction:** Two matrices can only be added or subtracted if and only if they have the same dimensions

**Matrix Multiplication:** Condition of matrix multiplication is, “Matrix multiplication between two matrices will be possible if and only if the number of columns of the first matrix is equal to the number of rows of the second matrix

**Idempotent Matrix:** The matrix A is idempotent matrix if and only if A.A=A

**Involuntary matrix:** If A.A = I or A^-1 = A then the matrix A is said to be an involuntary matrix.

**Nilpotent matrix:** A nilpotent matrix is a square matrix N such that A.A=0 for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.

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