Does the determinant of a substitution matrix need to be $pm 1$?Prove equivalence between the determinant of a matrix and the product of specific submatricesCalculate the determinant of given matrixdeterminant of matrix $X$Calculate the determinant of a matrix given a simple conditionDeterminant of an anti-diagonal block matrix$2times 2$ block Toeplitz determinantDeterminant of Skew-Symmetric MatricesDoes rotating a matrix change its determinant?Determinant of special partitioned matrix in terms of submatrix determinantsFind the determinant of the following $5times 5$ real matrix:
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Does the determinant of a substitution matrix need to be $pm 1$?
Prove equivalence between the determinant of a matrix and the product of specific submatricesCalculate the determinant of given matrixdeterminant of matrix $X$Calculate the determinant of a matrix given a simple conditionDeterminant of an anti-diagonal block matrix$2times 2$ block Toeplitz determinantDeterminant of Skew-Symmetric MatricesDoes rotating a matrix change its determinant?Determinant of special partitioned matrix in terms of submatrix determinantsFind the determinant of the following $5times 5$ real matrix:
$begingroup$
The substitutional rule for the Fibonacci sequence is $sigma: L rightarrow LS, S rightarrow L$, is:
$$
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
1 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LS \
S\
endarray right ), $$
and $mathrmdet, mathcalS = -1$.
The substitutional rule for the Octonacci (or Pell) sequence is $sigma: L rightarrow LLS, Srightarrow L$, or:
beginequation
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
2 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LLS \
S\
endarray right ).
endequation
with $mathrmdet, mathcalS = -1$.
The substitution rule for the Ammmann-Beenker tiling is:
$$
left ( beginarraycc
3 & 2 \
4 & 3
endarray right ) $$
whose determinant is $1$.
Do substitution matrices need to have determinant $pm 1$?
What is the physical meaning?
sequences-and-series matrices tiling
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
The substitutional rule for the Fibonacci sequence is $sigma: L rightarrow LS, S rightarrow L$, is:
$$
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
1 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LS \
S\
endarray right ), $$
and $mathrmdet, mathcalS = -1$.
The substitutional rule for the Octonacci (or Pell) sequence is $sigma: L rightarrow LLS, Srightarrow L$, or:
beginequation
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
2 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LLS \
S\
endarray right ).
endequation
with $mathrmdet, mathcalS = -1$.
The substitution rule for the Ammmann-Beenker tiling is:
$$
left ( beginarraycc
3 & 2 \
4 & 3
endarray right ) $$
whose determinant is $1$.
Do substitution matrices need to have determinant $pm 1$?
What is the physical meaning?
sequences-and-series matrices tiling
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
The substitutional rule for the Fibonacci sequence is $sigma: L rightarrow LS, S rightarrow L$, is:
$$
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
1 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LS \
S\
endarray right ), $$
and $mathrmdet, mathcalS = -1$.
The substitutional rule for the Octonacci (or Pell) sequence is $sigma: L rightarrow LLS, Srightarrow L$, or:
beginequation
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
2 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LLS \
S\
endarray right ).
endequation
with $mathrmdet, mathcalS = -1$.
The substitution rule for the Ammmann-Beenker tiling is:
$$
left ( beginarraycc
3 & 2 \
4 & 3
endarray right ) $$
whose determinant is $1$.
Do substitution matrices need to have determinant $pm 1$?
What is the physical meaning?
sequences-and-series matrices tiling
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
$endgroup$
The substitutional rule for the Fibonacci sequence is $sigma: L rightarrow LS, S rightarrow L$, is:
$$
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
1 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LS \
S\
endarray right ), $$
and $mathrmdet, mathcalS = -1$.
The substitutional rule for the Octonacci (or Pell) sequence is $sigma: L rightarrow LLS, Srightarrow L$, or:
beginequation
sigma : left ( beginarrayc
L \
S\
endarray right ) rightarrow underbraceleft ( beginarraycc
2 & 1 \
1 & 0 \
endarray right )_=mathcalS left ( beginarrayc
L \
S\
endarray right ) = left ( beginarrayc
LLS \
S\
endarray right ).
endequation
with $mathrmdet, mathcalS = -1$.
The substitution rule for the Ammmann-Beenker tiling is:
$$
left ( beginarraycc
3 & 2 \
4 & 3
endarray right ) $$
whose determinant is $1$.
Do substitution matrices need to have determinant $pm 1$?
What is the physical meaning?
sequences-and-series matrices tiling
sequences-and-series matrices tiling
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
edited Mar 29 at 11:45
J. W. Tanner
4,4791320
4,4791320
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
asked Mar 29 at 11:36
SuperCiociaSuperCiocia
295213
295213
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, for instance the Thue-Morse substitution $a mapsto ab, b mapsto ba$ has determinant $0$, and the period doubling substitution $a mapsto ab, b mapsto aa$ has determinant $-2$.
When a substitution matrix has determinant $pm 1$, then we call the substitution unimodular. You can find plenty of literature on unimodular Pisot substitutions for instance, which is related to the long standing open question known as the Pisot conjecture or Pisot substitution conjecture.
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
No, for instance the Thue-Morse substitution $a mapsto ab, b mapsto ba$ has determinant $0$, and the period doubling substitution $a mapsto ab, b mapsto aa$ has determinant $-2$.
When a substitution matrix has determinant $pm 1$, then we call the substitution unimodular. You can find plenty of literature on unimodular Pisot substitutions for instance, which is related to the long standing open question known as the Pisot conjecture or Pisot substitution conjecture.
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
$endgroup$
add a comment |
$begingroup$
No, for instance the Thue-Morse substitution $a mapsto ab, b mapsto ba$ has determinant $0$, and the period doubling substitution $a mapsto ab, b mapsto aa$ has determinant $-2$.
When a substitution matrix has determinant $pm 1$, then we call the substitution unimodular. You can find plenty of literature on unimodular Pisot substitutions for instance, which is related to the long standing open question known as the Pisot conjecture or Pisot substitution conjecture.
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
$endgroup$
add a comment |
$begingroup$
No, for instance the Thue-Morse substitution $a mapsto ab, b mapsto ba$ has determinant $0$, and the period doubling substitution $a mapsto ab, b mapsto aa$ has determinant $-2$.
When a substitution matrix has determinant $pm 1$, then we call the substitution unimodular. You can find plenty of literature on unimodular Pisot substitutions for instance, which is related to the long standing open question known as the Pisot conjecture or Pisot substitution conjecture.
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
$endgroup$
No, for instance the Thue-Morse substitution $a mapsto ab, b mapsto ba$ has determinant $0$, and the period doubling substitution $a mapsto ab, b mapsto aa$ has determinant $-2$.
When a substitution matrix has determinant $pm 1$, then we call the substitution unimodular. You can find plenty of literature on unimodular Pisot substitutions for instance, which is related to the long standing open question known as the Pisot conjecture or Pisot substitution conjecture.
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
edited Mar 29 at 13:33
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
answered Mar 29 at 13:23
Dan RustDan Rust
23k114984
23k114984
add a comment |
add a comment |
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