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Types of Matrices: Definition & Differences

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  • 0:03 Matrix
  • 0:36 Types of Matrices
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Instructor
Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

Matrices are rectangular arrays of numbers arranged in the form of rows and columns. In this lesson, you will explore the various types of matrices with examples of each.

Matrix

In order to arrange numerous numbers, mathematics provides a simple solution: matrices. A matrix can be defined as a rectangular grid of numbers, symbols, and expressions arranged in rows and columns. These grids are usually charted by brackets around them.

The dimensions of a matrix are represented as R X C, where R is the number of rows and C is the number of columns. This R X C notation is also called the order of the matrix.

Types of Matrices

There are various types of matrices, depending on their structure. Let's explore the most common types:

Null Matrix

A matrix that has all 0 elements is called a null matrix. It can be of any order. For example, we could have a null matrix of the order 2 X 3. It's also a singular matrix, since it does not have an inverse and its determinant is 0.


Null Matrix


Any matrix that does have an inverse can be called a regular matrix.

Row Matrix

A row matrix is a matrix with only one row. Its order would be 1 X C, where C is the number of columns. For example, here's a row matrix of the order 1 X 5:


Row Matrix


Column Matrix

A column matrix is a matrix with only one column. It is represented by an order of R X 1, where R is the number of rows. Here's a column matrix of the order 3 X 1:


Column Matrix


Square Matrix

A matrix where the number of rows is equal to the number of columns is called a square matrix. Here's a square matrix of the order 2 X 2:


Square Matrix


Diagonal Matrix

A diagonal matrix is a square matrix where all the elements are 0 except for those in the diagonal from the top left corner to the bottom right corner. Let's take a look at a diagonal matrix of order 4 X 4:


Diagonal Matrix


A special type of diagonal matrix, where all the diagonal elements are equal is called a scalar matrix. We can see a 3 X 3 scalar matrix here:


Scalar Matrix


A scalar matrix whose diagonal elements are all 1 is called a unit matrix, or identity matrix.


Unit Matrix


Upper Triangular Matrix

A square matrix where all the elements below the left-right diagonal are 0 is called an upper triangular matrix. Here's an upper triangular matrix of order 3 X 3:


Upper Triangular Matrix


Lower Triangular Matrix

A square matrix where all the elements above the left-right diagonal are 0 is called a lower triangular matrix. Here's what a lower triangular matrix of order 3 X 3 could look like:


Lower Triangular Matrix


Symmetric Matrix

A matrix whose transpose is the same as the original matrix is called a symmetric matrix. Only a square matrix can be a symmetric matrix. The transpose of a matrix is another matrix that is formed by switching the rows and columns of a given matrix. The given matrix A is a 3 X 3 symmetric matrix, since it's the same as its transpose AT.


Symmetric Matrix


Antisymmetric Matrix

A square matrix whose transpose is its negation is an antisymmetric matrix, or skew-symmetric matrix. The negation of a matrix is a matrix formed by negating the signs of all the entries:


Antisymmetric matrix


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Additional Activities

More Kinds of Matrices


Patterns

The identity matrix is a square matrix with ones on the diagonal.

Here is an example of a 3X3 identity matrix: ((1, 0, 0);(0, 1, 0);(0, 0, 1))

A diagonal matrix is a matrix with some elements in place of the ones in the identity matrix.

Can you give an example of a 3X3 diagonal matrix?

In a Toeplitz matrix, each descending diagonal from upper left to lower right has all the same elements along that diagonal. An example is:

((6, 7, 8, 9);(4, 6, 7, 8);(1, 4, 6, 7);(0, 1, 4, 6);(2, 0, 1, 4))

Is this matrix Toeplitz? ((6, 3, 8);(4, 9, 7);(1, 4, 6))

Graphs

The Adjacency matrix A, also known as the connection matrix, indicates which pairs of vertices are adjacent.

A(i,j) = 1 if vertices i and j have an edge between them. Otherwise the element is zero.

A graph of 5 vertices (or points) with all edges (or lines) connected is represented by:

((0,1,1,1,1);(1,0,1,1,1);(1,1,0,1,1);(1,1,1,0,1);(1,1,1,1,0))

Sparsity

Sparse matrices are condensed versions of very large matrices that are mostly zeros.

One simple method is the coordinate list (COO) format of (row, column, value) triple called tuple.

The 3X3 identity in COO format is ((1,1,1);(2, 2, 1);(3, 3, 1)).

Matrix within a Matrix

The Kronecker product of two matrices is a tensor matrix. If A is an mXn matrix and B is pXq, then their Kronecker product is an (mp) X ( nq) matrix equal to

((a(1, 1)B, a(1, 2)B);(a(2, 1)B, a(2, 2)B))

Note that A kron B is not equal to B kron A.

What is the Kronecker product of A kron B?

if A = ((1, 2);(3, 4);(1, 0)) and B = ((0, 5 ,2);(6, 7, 3))

====================

Answers to questions:

3X3 diagonal: ((1, 0, 0);(0, 6, 0);(0, 0, 4))

Toeplitz: No

A kron B: ((0, 5, 2, 0, 10, 4);(6, 7, 3, 12, 14, 6);(0, 15, 6, 0, 20, 8);(18, 21, 9, 24, 28, 12);(0, 5, 2, 0, 0, 0);(6, 7, 3, 0, 0, 0))

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