How to find the value of this determinant?
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I'm wondering how to find the value of this determinant.
$$left[ {begin{array}{*{20}{c}}
0&{{x_1}}&{{x_2}}&{{x_3}}& ddots &{{x_n}} \
{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}}& ddots \
{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}} \
{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}} \
ddots &{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}} \
{{x_n}}& ddots &{{x_3}}&{{x_2}}&{{x_1}}&0
end{array}} right]$$
I tried for some small $n$ and couldn't see any clue to find its value expressed by an easy formula. Any help will be appreciated.
linear-algebra matrices polynomials determinant multilinear-algebra
asked Nov 16 at 9:28
gžd15
650510
|
show 1 more comment
up vote
1
down vote
favorite
I'm wondering how to find the value of this determinant.
$$left[ {begin{array}{*{20}{c}}
0&{{x_1}}&{{x_2}}&{{x_3}}& ddots &{{x_n}} \
{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}}& ddots \
{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}} \
{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}} \
ddots &{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}} \
{{x_n}}& ddots &{{x_3}}&{{x_2}}&{{x_1}}&0
end{array}} right]$$
I tried for some small $n$ and couldn't see any clue to find its value expressed by an easy formula. Any help will be appreciated.
linear-algebra matrices polynomials determinant multilinear-algebra
asked Nov 16 at 9:28
gžd15
650510
Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
4
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm wondering how to find the value of this determinant.
$$left[ {begin{array}{*{20}{c}}
0&{{x_1}}&{{x_2}}&{{x_3}}& ddots &{{x_n}} \
{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}}& ddots \
{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}} \
{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}} \
ddots &{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}} \
{{x_n}}& ddots &{{x_3}}&{{x_2}}&{{x_1}}&0
end{array}} right]$$
I tried for some small $n$ and couldn't see any clue to find its value expressed by an easy formula. Any help will be appreciated.
linear-algebra matrices polynomials determinant multilinear-algebra
asked Nov 16 at 9:28
gžd15
650510
I'm wondering how to find the value of this determinant.
$$left[ {begin{array}{*{20}{c}}
0&{{x_1}}&{{x_2}}&{{x_3}}& ddots &{{x_n}} \
{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}}& ddots \
{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}} \
{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}} \
ddots &{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}} \
{{x_n}}& ddots &{{x_3}}&{{x_2}}&{{x_1}}&0
end{array}} right]$$
I tried for some small $n$ and couldn't see any clue to find its value expressed by an easy formula. Any help will be appreciated.
linear-algebra matrices polynomials determinant multilinear-algebra
linear-algebra matrices polynomials determinant multilinear-algebra
asked Nov 16 at 9:28
gžd15
650510
asked Nov 16 at 9:28
gžd15
650510
asked Nov 16 at 9:28
gžd15
650510
asked Nov 16 at 9:28
gžd15
650510
asked Nov 16 at 9:28
gžd15
650510
650510
Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
4
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43
|
show 1 more comment
Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
4
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43
Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
4
4
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43
|
show 1 more comment
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Is this matrix $6 times 6$ ? $ (n=5)?$
– Widawensen
Nov 16 at 9:49
Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result.
– Widawensen
Nov 16 at 9:59
see yutsumura.com/determinant-of-a-general-circulant-matrix
– Aleksas Domarkas
Nov 16 at 10:29
4
This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help.
– jobe
Nov 16 at 10:33
@Widawensen you are right
– Aleksas Domarkas
Nov 16 at 10:43