Types of Matrices with Examples
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
- π Square matrices have an equal number of rows and columns.
- π Rectangular matrices (non-square) have a different number of rows and columns.
- π’ Row matrices (or row vectors) consist of a single row with multiple columns.
- π Column matrices (or column vectors) consist of a single column with multiple rows.
- π© Identity matrices have ones on the main diagonal and zeros elsewhere.
- π« Zero matrices (null matrices) consist entirely of zeros.
- π Diagonal matrices have non-zero values on the main diagonal and zeros elsewhere.
- πΌ Scalar matrices are similar to identity matrices but with the same number on the main diagonal and zeros elsewhere.
- βΉ Lower triangular matrices have all entries above the main diagonal as zeros.
- βΊ Upper triangular matrices have all entries below the main diagonal as zeros.
- π The transpose of a matrix is formed by swapping its rows and columns.
- π€ Symmetric matrices are those where the matrix is equal to its transpose.
- π 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
π 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.
π’ 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.
πΌ 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.
π 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
Keywords
π‘Square Matrix
π‘Rectangular Matrix
π‘Row Matrix
π‘Column Matrix
π‘Identity Matrix
π‘Zero Matrix
π‘Diagonal Matrix
π‘Scalar Matrix
π‘Triangular Matrix
π‘Transpose of a Matrix
π‘Symmetric Matrix
π‘Skew Symmetric 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
hi guys in this video we'll know the
types of matrices
and here i have list the most important
matrices or most
important special matrices which are
square
rectangular row column identity
zero diagonal scalar lower triangular
upper triangular transpose symmetric
and skew symmetric matrix before to
introduce
this one we will know how to compute the
negative of
matrix first of these and
of course here we will list them as this
order
first one square matrix from the name
square matrix
as we know the square is
each opposite side are equal
this is the same thing so here the
number of rows
equal number of
column
here i mean here of this one number
of i mean of this hash
here number of and the same as this one
so here number of rows equal to number
of column four square matrix for example
the simple one is one by one matrix
for example here minus one
here one by one matrix this one
another example five
two three minus one
this two by two matrix and so on
we can put here three by three matrix
this three by three matrix i will
explain
more what is this one and what is this
one
here this one number of rows
one two
and here in purple
this one number of columns
the same thing as this one and of course
here you can put
any size as you wish for example
100 by 100 matrix which is square matrix
the second one is rectangular we will
write it here
here rectangular matrix opposite of this
one
number
of rows not equal
number this i mean number of
column
for example
this two by three matrix
two rows and three column
which is rectangular matrix
another example
this one 4 by 2
and so on you can write any size you
want
of course the number of rows not equal
to the number of columns then it will be
rectangular matrix the third one
is row matrix
roll matrix or raw vector
and from here the name row vector it's
mean
only one row so example of this one
it's very easy which is
like this one or minus one
two it has one row
just one row one by two this because one
row and two column
and here this one one by three
and so on because here three
three column and as you noticed here
here one here one so
it has only one row so this is
the row matrix now we will talk about
column matrix
column matrix or column vector for
example
it has only one column three
two this has only one column
so this one two rows and one column so
two by one
another example
this one three by one matrix
and so on
as you can see here it has only one
column
this one has only one column so this is
column vectors example for column vector
now we will know about identity matrix
identity matrix for example
one this one by one matrix
or two by two matrix
one zero zero one
as you can see here
as you can see here on the main diagonal
as ones
here's this called main diagonal
this is main diagonal here one one the
entries
one one and the entries two two
of course this one two by two matrix
and here an identity matrix has ones on
the main diagonals and everywhere else
are zeros so here zeros let's introduce
three by three matrix
this is three by three matrix
as you can see here on the main diagonal
once
and everywhere else are zeros
and so on you can put 10 by 10 matrix
which has the same thing on the main
diagonal ones and
everywhere else are zeros now we will
introduce
the zero matrix
here zero matrix or they called it null
matrix
which is very simple
zero here only zero this is one by one
matrix
it's opposite of this one here only one
and here
only zero or you can write
zero zero zero zero this two by two
matrix
and they called it null matrix or zero
matrix
you can put two by three three by two
three by three or any size you wish
now we will talk about diagonal matrix
diagonal matrix diagonal matrix it's
similar to identity matrix
but in diagonal matrix has numbers here
not equal to zero so
for example here
two zero zero minus one
you can see here on the main diagonal
numbers and everywhere else are zero
so a diagonal matrix has zero
everywhere except the main diagonal
this two by two matrix let's put example
for
three by three matrix
as you can see here on the main diagonal
here
has number everywhere else are zero this
is 3 by
3 matrix and so on
now we will introduce the scalar matrix
example of scalar matrix
scalar matrix has similar to identity
matrix
but here in scalar matrix has numbers
on the main diagonal and they are the
same
so a scalar matrix has all main diagonal
entries the same with 0 everywhere else
so here 0 and here the same numbers this
is two by two matrix
another example
three zero zero three
this is the scalar matrix because they
have the same number and
everywhere else are zero this is the
same two by two matrix
for three by three matrix we will put
example here
three by three matrix
so the main diagonal here has the same
number
eight eight eight so this is scalar
matrix
and so on now we'll introduce
triangular matrix
we have triangular matrix and here
triangular matrix has
either lower triangular matrix or
upper triangular matrix you can say
this one is triangular matrix but if you
want to be more specific you can say
lower triangular matrix or upper
triangular matrix what is the lower
triangular matrix for example
let's look like this for example here
5 zero
three minus two this is
two by two lower triangular
why it's triangular matrix because here
we have right triangle
you can see here we have right triangle
and
here if this shape like this
then it is lower triangular matrix
another example for three by three
matrix
this is three by three matrix
and as you can see here
right triangle look like this
here and so on
this is lower triangular matrix as you
can see
here the right angle here right angle
here
for upper triangular matrix this angle
it's opposite it will be here for
example here
look like this
2 5 0
minus 5 for example
see right triangle look like this
and here the right angle here so
that's why this called upper triangular
matrix
another example three by three matrix
the right triangle
here it has here 90 degree
and so on
lower triangular matrix is when all
entries above the main diagonal
zero above the main diagonal zero
where upper triangular matrix
is when the all entries below
the main diagonal is zero so here
below the main diagonal are zeros
now we will introduce for you transpose
of matrix
transpose of matrix is very simple
it's only you have to swap
the rows and column so rows
become columns
and columns
become rows
it's very simple here for example
let's we have a equal
so here the a transpose the symbol of
transpose here let's look like this so
a transpose this is transpose here t
equal we have to swap the rows
with column so here this row
it will become column so five
two i swap this one to here
the same thing here this row it will
become column so
negative three four
this row become column so zero
one
another example
b equal i'll put symbol example here
5 2 9.
this one by three
because one row by three column
b transpose
it will we have to swap this
row to become column so it will be like
this
five two
nine so now we have
three by one
the same here we have three rows and two
columns
three rows two column then
after the transpose this one
transformed to 2 by 3
so a 3 by 2 transform
to this matrix which is
two by three so a transpose two by three
now we will talk about the negative of
matrix
here it's very simple the negative of
matrix
for example if we have
let's say a equal
5 to zero
minus three
then the negative of a
the negative of matrix a equal
we have only two multiply five multiply
by negative one which is
minus five multiply by negative one
which is minus two this one it
will remain the same minus one
multiply by minus three it will be three
now we will know about symmetric matrix
here we have symmetric matrix which is
a equal a transpose so if this condition
satisfy
then we can say this is symmetric matrix
so a matrix whose transpose is the same
as the original matrix for example
a this is given matrix
so if we want to test if this symmetric
or not
we have to take the transpose so let's
write it in another color
so then a transpose
equal
so this one it will become column one
two three this route will become the
column
two minus five
minus one this row the third row
will be the third column so three
negative one six
let's compare these two one one
two two three three two two
negative five negative five negative one
negative one
three three minus one minus one
six six since these matrices
are equal then or
since
a equal a transpose
then a
is symmetric matrix
now we will introduce for you the
skew symmetric matrix
here we have skew symmetric matrix
it is skew symmetric matrix if satisfy
this condition
a transpose equal to the negative of a
this mean
a matrix whose transpose is equal to the
negative of the original matrix
example for this one
let's say we have this a
to test if this skew symmetric or not
we have to compute a transpose and the
negative of a
so here we will compute this one first
a transpose
this row to become column zero
six negative three
and this row to become column negative
six
zero negative one the last row
it will become the last column here
three one
zero
now we will compute the negative of a
the negative of a it's very simple we
have to compute
the negative of this one so it's only
multiplied by negative 1
for the whole matrix so here zero
minus six
three six
zero one
negative three
negative one
zero
now we have to compare this one
with this one so zero zero negative six
negative six
three three six six zero
zero one one negative three negative
three
negative one negative one zero zero
so since
this matrix equal to this matrix then
it is skew symmetric matrix
since a transpose equal
negative of a then this
a is
skew
symmetric
matrix
that's it so we have left the most
important
matrices in linear algebra and it's very
important to distinguish between them if
you have
any question feel free to put comment
on this video see you in the next video
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