Types of Matrices with Examples

Mulkek
30 Sept 202023:37
EducationalLearning
32 Likes 10 Comments

TLDRThis video script introduces various types of matrices in linear algebra, including square, rectangular, row, column, identity, zero, diagonal, scalar, lower triangular, upper triangular, transpose, negative, symmetric, and skew symmetric matrices. It explains the characteristics of each matrix type, using examples to illustrate their properties and how to compute them, such as transposing a matrix or finding the negative of a matrix. The video aims to help viewers understand the differences between these matrix types and their applications in linear algebra.

Takeaways
  • πŸ“Š Square matrices have an equal number of rows and columns.
  • πŸ“ Rectangular matrices (non-square) have a different number of rows and columns.
  • πŸ”’ Row matrices (or row vectors) consist of a single row with multiple columns.
  • πŸ”„ Column matrices (or column vectors) consist of a single column with multiple rows.
  • 🎩 Identity matrices have ones on the main diagonal and zeros elsewhere.
  • 🚫 Zero matrices (null matrices) consist entirely of zeros.
  • πŸ”— Diagonal matrices have non-zero values on the main diagonal and zeros elsewhere.
  • πŸ”Ό Scalar matrices are similar to identity matrices but with the same number on the main diagonal and zeros elsewhere.
  • ⏹ Lower triangular matrices have all entries above the main diagonal as zeros.
  • ⏺ Upper triangular matrices have all entries below the main diagonal as zeros.
  • πŸ”„ The transpose of a matrix is formed by swapping its rows and columns.
  • 🀝 Symmetric matrices are those where the matrix is equal to its transpose.
  • πŸ’” Skew-symmetric matrices are those where the transpose is equal to the negative of the original matrix.
Q & A
  • What is the primary characteristic of a square matrix?

    -A square matrix is characterized by having an equal number of rows and columns. For instance, a 4x4 matrix has four rows and four columns.

  • How can you identify a rectangular matrix?

    -A rectangular matrix is identified by having a different number of rows compared to columns. For example, a 2x3 matrix has two rows and three columns.

  • What defines a row matrix or row vector?

    -A row matrix or row vector is defined by having only one row. The number of columns can vary, such as in a 1x2 matrix or 1x3 matrix.

  • What is the defining feature of a column matrix or column vector?

    -A column matrix or column vector is defined by having only one column. The number of rows can vary, such as in a 2x1 matrix or 3x1 matrix.

  • How can you recognize an identity matrix?

    -An identity matrix is recognized by having ones on the main diagonal and zeros elsewhere. The size of the identity matrix can be any square shape, like 1x1, 2x2, or 3x3, etc.

  • What makes a matrix a zero matrix or null matrix?

    -A zero matrix or null matrix is characterized by having all its elements as zero. The size of the zero matrix can also vary, such as 1x1, 2x2, or 3x3, etc.

  • What is the difference between a diagonal matrix and a scalar matrix?

    -A diagonal matrix has non-zero numbers on the main diagonal with zeros elsewhere, while a scalar matrix has the same number on the main diagonal with zeros elsewhere.

  • How can you determine if a matrix is lower triangular?

    -A matrix is lower triangular if all entries above the main diagonal are zero. The shape of the matrix forms a right triangle when viewed from the perspective of the main diagonal.

  • What is the definition of an upper triangular matrix?

    -An upper triangular matrix is defined by having all entries below the main diagonal as zero. The shape of the matrix forms a right triangle when viewed from the perspective of the main diagonal, but with the triangle pointing upwards.

  • What does it mean to transpose a matrix?

    -To transpose a matrix means to swap its rows with columns. Each element in the new matrix will be found in the position that was directly across the original matrix's main diagonal.

  • How can you identify a symmetric matrix?

    -A symmetric matrix is identified by being equal to its transpose. In other words, the matrix remains the same when its rows and columns are swapped.

  • What condition must a matrix satisfy to be considered skew-symmetric?

    -A matrix is considered skew-symmetric if its transpose is equal to the negative of the original matrix. This means that the matrix remains the same when its rows and columns are swapped, but with every element negated.

Outlines
00:00
πŸ“Š Introduction to Matrix Types

This paragraph introduces the viewer to various types of matrices, emphasizing the importance of understanding these mathematical structures. It begins by listing the different types of matrices such as square, rectangular, row, column, identity, zero, diagonal, scalar, lower triangular, upper triangular, transpose, symmetric, and skew symmetric matrices. The explanation starts with the square matrix, detailing its properties where the number of rows equals the number of columns, and provides examples ranging from 1x1 to larger matrices. The paragraph also touches on rectangular matrices, which have an unequal number of rows and columns, and row and column matrices, also known as vectors, which have only one row or column respectively.

05:03
πŸ”’ Special Matrices: Identity, Zero, Diagonal, and Scalar

This section delves into four special types of matrices: identity, zero, diagonal, and scalar. The identity matrix is characterized by ones on the main diagonal and zeros elsewhere, with examples provided for 1x1, 2x2, and larger matrices. The zero or null matrix is composed entirely of zeros. The diagonal matrix is similar to the identity matrix but with non-zero entries on the diagonal. Scalar matrices are like diagonal matrices but with all diagonal entries being the same number. The explanation includes the properties and examples of 2x2 and 3x3 matrices for each type, highlighting their unique characteristics and uses in linear algebra.

10:07
πŸ”Ό Understanding Triangular, Transpose, and Negative Matrices

This paragraph focuses on triangular matrices, which are divided into lower and upper triangular matrices based on the position of non-zero entries relative to the diagonal. Lower triangular matrices have non-zero entries below the diagonal, while upper triangular matrices have non-zero entries above the diagonal. The concept of the transpose of a matrix is introduced, explaining that it involves swapping rows with columns. Several examples are provided to illustrate the process of transposing matrices of different sizes. The negative of a matrix is also explained, which involves multiplying each element by -1, with an example demonstrating how to calculate the negative of a given matrix.

15:10
πŸ”„ Symmetric and Skew Symmetric Matrices

The final paragraph discusses symmetric and skew symmetric matrices. A symmetric matrix is one where the original matrix equals its transpose, meaning the matrix remains unchanged when its rows and columns are interchanged. The explanation includes the process of verifying if a matrix is symmetric by comparing the original matrix with its transpose. A skew symmetric matrix is defined as one whose transpose is equal to the negative of the original matrix. The process of determining if a matrix is skew symmetric is explained with an example, showing the computation of the transpose and the negative of the matrix, and then comparing them to ascertain skew symmetry.

Mindmap
Upper Triangular Matrix
Lower Triangular Matrix
Skew Symmetric Matrix
Symmetric Matrix
Negative of a Matrix
Transpose of a Matrix
Triangular Matrix
Scalar Matrix
Diagonal Matrix
Zero Matrix (Null Matrix)
Identity Matrix
Column Matrix (Vector)
Row Matrix (Vector)
Rectangular Matrix
Square Matrix
Special Matrices
Matrix Fundamentals
Types of Matrices in Linear Algebra
Alert
Keywords
πŸ’‘Square Matrix
A square matrix is a matrix with the same number of rows and columns. It is a fundamental concept in linear algebra and has various applications in solving systems of equations, matrix multiplication, and eigenvalue calculations. In the video, examples of square matrices include a 1x1 matrix with a value of -1 and a 2x2 matrix with elements 5, 0, -3, and 1.
πŸ’‘Rectangular Matrix
A rectangular matrix, also known as a non-square matrix, has a different number of rows and columns. This type of matrix is common in representing systems of linear equations where the number of variables is not equal to the number of equations. In the video, examples include a 2x3 matrix and a 4x2 matrix, illustrating the difference in dimensions.
πŸ’‘Row Matrix
A row matrix, or row vector, is a matrix with a single row and multiple columns. It represents a set of data points in a linear algebraic context and is often used in operations such as dot products and matrix multiplication. The video provides examples of row matrices with varying numbers of columns, such as a 1x2 matrix with -1, 2, and a 1x3 matrix with elements 3, -1, 2.
πŸ’‘Column Matrix
A column matrix, or column vector, is a matrix with a single column and multiple rows. It is used in similar contexts as row matrices, including linear transformations and representing vectors in a vector space. The video provides examples of column matrices with varying numbers of rows, such as a 2x1 matrix with 3, 2 and a 3x1 matrix with elements 5, -3, 1.
πŸ’‘Identity Matrix
An identity matrix is a special square matrix with ones on the main diagonal and zeros elsewhere. It plays a crucial role in linear algebra as it serves as the multiplicative identity, meaning any matrix multiplied by the identity matrix results in the original matrix. The video explains that an identity matrix can be of any size, with examples including 1x1 and 2x2 matrices.
πŸ’‘Zero Matrix
A zero matrix, also known as a null matrix, is a matrix where all elements are zero. It represents the additive identity in matrix operations, as adding a zero matrix to any matrix results in the original matrix. The video describes zero matrices of different sizes, including 1x1 and 2x2 matrices.
πŸ’‘Diagonal Matrix
A diagonal matrix is a square matrix with non-zero entries on the main diagonal and zero entries elsewhere. This type of matrix simplifies many matrix operations and has applications in various fields, including physics and engineering. The video provides examples of diagonal matrices, such as a 2x2 matrix with 2, 0, 0, -1.
πŸ’‘Scalar Matrix
A scalar matrix is a diagonal matrix with all diagonal entries being the same scalar value. It scales the basis vectors of a vector space by that scalar value without changing the span or direction of the vectors. The video explains that a scalar matrix has identical values on the main diagonal and zeros elsewhere, with examples including a 2x2 matrix with a single value 0 in the diagonal.
πŸ’‘Triangular Matrix
A triangular matrix is a square matrix with all entries above the main diagonal being zero (lower triangular) or all entries below the main diagonal being zero (upper triangular). These matrices have simplified forms that make certain operations, such as inversion and determinant calculation, more straightforward. The video provides examples of both lower and upper triangular matrices, such as a 2x2 matrix with 5, 0, -3, and 2.
πŸ’‘Transpose of a Matrix
The transpose of a matrix is a new matrix obtained by interchanging the rows and columns of the original matrix. This operation is fundamental in various areas of linear algebra, including solving systems of equations and understanding the properties of matrices. The video demonstrates the process of transposing a matrix, such as a 1x3 matrix becoming a 3x1 matrix after transposition.
πŸ’‘Symmetric Matrix
A symmetric matrix is a square matrix that is equal to its own transpose. This property means that the matrix is invariant under the operation of transposition, which is significant in various applications, such as representing real-valued functions in Fourier analysis. The video explains how to test for symmetry by comparing the original matrix with its transpose, with an example of a 3x3 matrix being symmetric because they are equal.
πŸ’‘Skew Symmetric Matrix
A skew symmetric matrix is a square matrix that is equal to the negative of its own transpose. This type of matrix has important applications in physics, particularly in the representation of cross products and angular momentum. The video describes how to determine if a matrix is skew symmetric by comparing the original matrix with the negative of its transpose, with an example of a 3x3 matrix being skew symmetric because its transpose equals the negative of the original matrix.
Highlights

The introduction of the concept of matrices, which are fundamental in linear algebra.

The explanation of square matrices, where the number of rows equals the number of columns.

Rectangular matrices are introduced, highlighting that their rows and columns are not equal.

The concept of row matrices or row vectors, which consist of a single row.

Column matrices or column vectors are explained, which consist of a single column.

Identity matrices are defined, characterized by ones on the main diagonal and zeros elsewhere.

Zero matrices, also known as null matrices, are introduced, consisting entirely of zeros.

Diagonal matrices are explained, which have non-zero entries on the main diagonal and zeros elsewhere.

Scalar matrices are introduced, a special type of diagonal matrix where all diagonal entries are the same.

Lower triangular matrices are defined, with all entries above the main diagonal being zero.

Upper triangular matrices are described, with all entries below the main diagonal being zero.

The concept of matrix transpose is explained, which involves swapping rows and columns.

Negative of a matrix is defined, which involves multiplying each element by negative one.

Symmetric matrices are introduced, which are equal to their own transpose.

Skew-symmetric matrices are defined, which are equal to the negative of their own transpose.

The importance of distinguishing between different types of matrices in linear algebra is emphasized.

The video offers an invitation for questions and comments, encouraging viewer interaction.

Transcripts
00:00

hi guys in this video we'll know the

00:03

types of matrices

00:04

and here i have list the most important

00:07

matrices or most

00:08

important special matrices which are

00:11

square

00:12

rectangular row column identity

00:15

zero diagonal scalar lower triangular

00:19

upper triangular transpose symmetric

00:22

and skew symmetric matrix before to

00:25

introduce

00:26

this one we will know how to compute the

00:28

negative of

00:30

matrix first of these and

00:33

of course here we will list them as this

00:36

order

00:37

first one square matrix from the name

00:40

square matrix

00:42

as we know the square is

00:45

each opposite side are equal

00:48

this is the same thing so here the

00:52

number of rows

00:59

equal number of

01:03

column

01:08

here i mean here of this one number

01:15

of i mean of this hash

01:18

here number of and the same as this one

01:22

so here number of rows equal to number

01:25

of column four square matrix for example

01:29

the simple one is one by one matrix

01:33

for example here minus one

01:37

here one by one matrix this one

01:40

another example five

01:43

two three minus one

01:47

this two by two matrix and so on

01:50

we can put here three by three matrix

01:57

this three by three matrix i will

01:59

explain

02:00

more what is this one and what is this

02:03

one

02:03

here this one number of rows

02:07

one two

02:10

and here in purple

02:14

this one number of columns

02:19

the same thing as this one and of course

02:22

here you can put

02:24

any size as you wish for example

02:28

100 by 100 matrix which is square matrix

02:31

the second one is rectangular we will

02:34

write it here

02:40

here rectangular matrix opposite of this

02:43

one

02:44

number

02:47

of rows not equal

02:52

number this i mean number of

02:56

column

03:01

for example

03:07

this two by three matrix

03:10

two rows and three column

03:14

which is rectangular matrix

03:18

another example

03:24

this one 4 by 2

03:27

and so on you can write any size you

03:30

want

03:31

of course the number of rows not equal

03:34

to the number of columns then it will be

03:36

rectangular matrix the third one

03:39

is row matrix

03:48

roll matrix or raw vector

03:51

and from here the name row vector it's

03:54

mean

03:54

only one row so example of this one

03:59

it's very easy which is

04:02

like this one or minus one

04:05

two it has one row

04:09

just one row one by two this because one

04:12

row and two column

04:17

and here this one one by three

04:22

and so on because here three

04:25

three column and as you noticed here

04:30

here one here one so

04:33

it has only one row so this is

04:37

the row matrix now we will talk about

04:40

column matrix

04:48

column matrix or column vector for

04:50

example

04:55

it has only one column three

04:59

two this has only one column

05:03

so this one two rows and one column so

05:06

two by one

05:10

another example

05:14

this one three by one matrix

05:17

and so on

05:22

as you can see here it has only one

05:24

column

05:30

this one has only one column so this is

05:33

column vectors example for column vector

05:36

now we will know about identity matrix

05:46

identity matrix for example

05:54

one this one by one matrix

05:59

or two by two matrix

06:03

one zero zero one

06:08

as you can see here

06:11

as you can see here on the main diagonal

06:13

as ones

06:15

here's this called main diagonal

06:32

this is main diagonal here one one the

06:35

entries

06:36

one one and the entries two two

06:40

of course this one two by two matrix

06:49

and here an identity matrix has ones on

06:54

the main diagonals and everywhere else

06:57

are zeros so here zeros let's introduce

07:02

three by three matrix

07:09

this is three by three matrix

07:13

as you can see here on the main diagonal

07:15

once

07:16

and everywhere else are zeros

07:19

and so on you can put 10 by 10 matrix

07:23

which has the same thing on the main

07:25

diagonal ones and

07:27

everywhere else are zeros now we will

07:30

introduce

07:31

the zero matrix

07:42

here zero matrix or they called it null

07:45

matrix

07:46

which is very simple

07:50

zero here only zero this is one by one

07:54

matrix

07:55

it's opposite of this one here only one

07:57

and here

07:58

only zero or you can write

08:02

zero zero zero zero this two by two

08:05

matrix

08:08

and they called it null matrix or zero

08:10

matrix

08:11

you can put two by three three by two

08:14

three by three or any size you wish

08:16

now we will talk about diagonal matrix

08:24

diagonal matrix diagonal matrix it's

08:26

similar to identity matrix

08:28

but in diagonal matrix has numbers here

08:32

not equal to zero so

08:35

for example here

08:42

two zero zero minus one

08:47

you can see here on the main diagonal

08:54

numbers and everywhere else are zero

08:58

so a diagonal matrix has zero

09:01

everywhere except the main diagonal

09:04

this two by two matrix let's put example

09:08

for

09:08

three by three matrix

09:15

as you can see here on the main diagonal

09:17

here

09:18

has number everywhere else are zero this

09:21

is 3 by

09:22

3 matrix and so on

09:26

now we will introduce the scalar matrix

09:34

example of scalar matrix

09:40

scalar matrix has similar to identity

09:43

matrix

09:44

but here in scalar matrix has numbers

09:48

on the main diagonal and they are the

09:51

same

09:52

so a scalar matrix has all main diagonal

09:56

entries the same with 0 everywhere else

09:59

so here 0 and here the same numbers this

10:03

is two by two matrix

10:07

another example

10:10

three zero zero three

10:15

this is the scalar matrix because they

10:17

have the same number and

10:18

everywhere else are zero this is the

10:21

same two by two matrix

10:25

for three by three matrix we will put

10:27

example here

10:31

three by three matrix

10:36

so the main diagonal here has the same

10:38

number

10:39

eight eight eight so this is scalar

10:43

matrix

10:44

and so on now we'll introduce

10:48

triangular matrix

11:00

we have triangular matrix and here

11:02

triangular matrix has

11:04

either lower triangular matrix or

11:07

upper triangular matrix you can say

11:10

this one is triangular matrix but if you

11:13

want to be more specific you can say

11:16

lower triangular matrix or upper

11:17

triangular matrix what is the lower

11:19

triangular matrix for example

11:22

let's look like this for example here

11:25

5 zero

11:29

three minus two this is

11:33

two by two lower triangular

11:36

why it's triangular matrix because here

11:41

we have right triangle

11:45

you can see here we have right triangle

11:48

and

11:48

here if this shape like this

11:52

then it is lower triangular matrix

11:55

another example for three by three

11:57

matrix

12:02

this is three by three matrix

12:07

and as you can see here

12:13

right triangle look like this

12:16

here and so on

12:21

this is lower triangular matrix as you

12:24

can see

12:24

here the right angle here right angle

12:27

here

12:28

for upper triangular matrix this angle

12:32

it's opposite it will be here for

12:35

example here

12:38

look like this

12:41

2 5 0

12:44

minus 5 for example

12:50

see right triangle look like this

12:56

and here the right angle here so

12:59

that's why this called upper triangular

13:02

matrix

13:05

another example three by three matrix

13:13

the right triangle

13:19

here it has here 90 degree

13:24

and so on

13:29

lower triangular matrix is when all

13:31

entries above the main diagonal

13:34

zero above the main diagonal zero

13:38

where upper triangular matrix

13:41

is when the all entries below

13:44

the main diagonal is zero so here

13:49

below the main diagonal are zeros

13:52

now we will introduce for you transpose

13:56

of matrix

14:04

transpose of matrix is very simple

14:08

it's only you have to swap

14:11

the rows and column so rows

14:15

become columns

14:21

and columns

14:27

become rows

14:32

it's very simple here for example

14:38

let's we have a equal

14:45

so here the a transpose the symbol of

14:48

transpose here let's look like this so

14:51

a transpose this is transpose here t

14:57

equal we have to swap the rows

15:00

with column so here this row

15:03

it will become column so five

15:07

two i swap this one to here

15:10

the same thing here this row it will

15:12

become column so

15:13

negative three four

15:17

this row become column so zero

15:20

one

15:24

another example

15:29

b equal i'll put symbol example here

15:35

5 2 9.

15:40

this one by three

15:44

because one row by three column

15:48

b transpose

15:52

it will we have to swap this

15:55

row to become column so it will be like

15:59

this

16:02

five two

16:06

nine so now we have

16:09

three by one

16:13

the same here we have three rows and two

16:16

columns

16:17

three rows two column then

16:20

after the transpose this one

16:24

transformed to 2 by 3

16:27

so a 3 by 2 transform

16:32

to this matrix which is

16:35

two by three so a transpose two by three

16:39

now we will talk about the negative of

16:42

matrix

16:48

here it's very simple the negative of

16:50

matrix

16:51

for example if we have

16:56

let's say a equal

17:02

5 to zero

17:06

minus three

17:12

then the negative of a

17:16

the negative of matrix a equal

17:20

we have only two multiply five multiply

17:23

by negative one which is

17:25

minus five multiply by negative one

17:29

which is minus two this one it

17:33

will remain the same minus one

17:36

multiply by minus three it will be three

17:42

now we will know about symmetric matrix

17:53

here we have symmetric matrix which is

17:56

a equal a transpose so if this condition

17:59

satisfy

18:00

then we can say this is symmetric matrix

18:04

so a matrix whose transpose is the same

18:07

as the original matrix for example

18:13

a this is given matrix

18:21

so if we want to test if this symmetric

18:24

or not

18:25

we have to take the transpose so let's

18:28

write it in another color

18:32

so then a transpose

18:35

equal

18:39

so this one it will become column one

18:43

two three this route will become the

18:47

column

18:48

two minus five

18:52

minus one this row the third row

18:55

will be the third column so three

18:59

negative one six

19:04

let's compare these two one one

19:07

two two three three two two

19:10

negative five negative five negative one

19:13

negative one

19:14

three three minus one minus one

19:17

six six since these matrices

19:20

are equal then or

19:24

since

19:27

a equal a transpose

19:32

then a

19:35

is symmetric matrix

19:51

now we will introduce for you the

19:54

skew symmetric matrix

20:05

here we have skew symmetric matrix

20:08

it is skew symmetric matrix if satisfy

20:11

this condition

20:12

a transpose equal to the negative of a

20:15

this mean

20:16

a matrix whose transpose is equal to the

20:20

negative of the original matrix

20:23

example for this one

20:28

let's say we have this a

20:38

to test if this skew symmetric or not

20:41

we have to compute a transpose and the

20:44

negative of a

20:46

so here we will compute this one first

20:49

a transpose

20:57

this row to become column zero

21:02

six negative three

21:05

and this row to become column negative

21:08

six

21:11

zero negative one the last row

21:17

it will become the last column here

21:20

three one

21:24

zero

21:27

now we will compute the negative of a

21:34

the negative of a it's very simple we

21:37

have to compute

21:39

the negative of this one so it's only

21:41

multiplied by negative 1

21:42

for the whole matrix so here zero

21:47

minus six

21:51

three six

21:56

zero one

22:01

negative three

22:05

negative one

22:10

zero

22:14

now we have to compare this one

22:18

with this one so zero zero negative six

22:21

negative six

22:22

three three six six zero

22:26

zero one one negative three negative

22:29

three

22:30

negative one negative one zero zero

22:33

so since

22:38

this matrix equal to this matrix then

22:41

it is skew symmetric matrix

22:45

since a transpose equal

22:49

negative of a then this

22:54

a is

22:58

skew

23:06

symmetric

23:11

matrix

23:19

that's it so we have left the most

23:22

important

23:23

matrices in linear algebra and it's very

23:26

important to distinguish between them if

23:28

you have

23:29

any question feel free to put comment

23:32

on this video see you in the next video