Time Complexity in finding the determinant of upper triangular matrix of order $n*n$
$begingroup$
I am trying to prove that the time Complexity in finding the determinant of upper triangular matrix of order $n*n$ is $O (n)$
My Approach:
Let us proceed this problem by taking an upper triangular matrix of order $3*3$
Let $A_3=
begin{bmatrix}
a_{11} & a_{12} & a_{13} \
0 & a_{22} & a_{23} \
0 & 0 & a_{33}
end{bmatrix}
$
Then, $det(A_3)=a_{11}*a_{22}*a_{33}$
$implies $ two multiplications
So, for an upper triangular matrix of order $n*n$ , we require $(n-1)$ multiplications
Now,
we know: Time Complexity $[f(n)]$ is defined as total number of multiplications required,i.e,
$f(n)=$ Total number of multiplications
And Big-O notation is the highest degree(i.e, if $f(n)=3n^2 +2n+1 $ ,then in Big-O notation it is represented by $O(n^2)$)
So,
here $(n-1)$ multiplication is required
$implies f(n)=n-1$
$implies O(n)$
Although my answer is right but i don't know whether i have proceeded correctly,whether i have used the concepts correctly or not...
So, anyone please verify it
linear-algebra determinant computational-complexity
$endgroup$
add a comment |
$begingroup$
I am trying to prove that the time Complexity in finding the determinant of upper triangular matrix of order $n*n$ is $O (n)$
My Approach:
Let us proceed this problem by taking an upper triangular matrix of order $3*3$
Let $A_3=
begin{bmatrix}
a_{11} & a_{12} & a_{13} \
0 & a_{22} & a_{23} \
0 & 0 & a_{33}
end{bmatrix}
$
Then, $det(A_3)=a_{11}*a_{22}*a_{33}$
$implies $ two multiplications
So, for an upper triangular matrix of order $n*n$ , we require $(n-1)$ multiplications
Now,
we know: Time Complexity $[f(n)]$ is defined as total number of multiplications required,i.e,
$f(n)=$ Total number of multiplications
And Big-O notation is the highest degree(i.e, if $f(n)=3n^2 +2n+1 $ ,then in Big-O notation it is represented by $O(n^2)$)
So,
here $(n-1)$ multiplication is required
$implies f(n)=n-1$
$implies O(n)$
Although my answer is right but i don't know whether i have proceeded correctly,whether i have used the concepts correctly or not...
So, anyone please verify it
linear-algebra determinant computational-complexity
$endgroup$
1
$begingroup$
Looks okay to me
$endgroup$
– Shubham Johri
Dec 17 '18 at 13:29
add a comment |
$begingroup$
I am trying to prove that the time Complexity in finding the determinant of upper triangular matrix of order $n*n$ is $O (n)$
My Approach:
Let us proceed this problem by taking an upper triangular matrix of order $3*3$
Let $A_3=
begin{bmatrix}
a_{11} & a_{12} & a_{13} \
0 & a_{22} & a_{23} \
0 & 0 & a_{33}
end{bmatrix}
$
Then, $det(A_3)=a_{11}*a_{22}*a_{33}$
$implies $ two multiplications
So, for an upper triangular matrix of order $n*n$ , we require $(n-1)$ multiplications
Now,
we know: Time Complexity $[f(n)]$ is defined as total number of multiplications required,i.e,
$f(n)=$ Total number of multiplications
And Big-O notation is the highest degree(i.e, if $f(n)=3n^2 +2n+1 $ ,then in Big-O notation it is represented by $O(n^2)$)
So,
here $(n-1)$ multiplication is required
$implies f(n)=n-1$
$implies O(n)$
Although my answer is right but i don't know whether i have proceeded correctly,whether i have used the concepts correctly or not...
So, anyone please verify it
linear-algebra determinant computational-complexity
$endgroup$
I am trying to prove that the time Complexity in finding the determinant of upper triangular matrix of order $n*n$ is $O (n)$
My Approach:
Let us proceed this problem by taking an upper triangular matrix of order $3*3$
Let $A_3=
begin{bmatrix}
a_{11} & a_{12} & a_{13} \
0 & a_{22} & a_{23} \
0 & 0 & a_{33}
end{bmatrix}
$
Then, $det(A_3)=a_{11}*a_{22}*a_{33}$
$implies $ two multiplications
So, for an upper triangular matrix of order $n*n$ , we require $(n-1)$ multiplications
Now,
we know: Time Complexity $[f(n)]$ is defined as total number of multiplications required,i.e,
$f(n)=$ Total number of multiplications
And Big-O notation is the highest degree(i.e, if $f(n)=3n^2 +2n+1 $ ,then in Big-O notation it is represented by $O(n^2)$)
So,
here $(n-1)$ multiplication is required
$implies f(n)=n-1$
$implies O(n)$
Although my answer is right but i don't know whether i have proceeded correctly,whether i have used the concepts correctly or not...
So, anyone please verify it
linear-algebra determinant computational-complexity
linear-algebra determinant computational-complexity
asked Dec 17 '18 at 13:26
SureshSuresh
309110
309110
1
$begingroup$
Looks okay to me
$endgroup$
– Shubham Johri
Dec 17 '18 at 13:29
add a comment |
1
$begingroup$
Looks okay to me
$endgroup$
– Shubham Johri
Dec 17 '18 at 13:29
1
1
$begingroup$
Looks okay to me
$endgroup$
– Shubham Johri
Dec 17 '18 at 13:29
$begingroup$
Looks okay to me
$endgroup$
– Shubham Johri
Dec 17 '18 at 13:29
add a comment |
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$begingroup$
Looks okay to me
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– Shubham Johri
Dec 17 '18 at 13:29