On the Smith normal form of skew E-W matrices

Armario, J. A.

doi:10.1080/03081087.2016.1186148

Cited by 1 publication

(7 citation statements)

References 11 publications

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“…Remark The equality

b_{2 t + 1} = 1

implies that

b_{1} = \dots = b_{2 t} = 1

. Thus it follows that

s_{2} = \dots = s_{2 t + 1} = 2

, which was already shown in , Theorem 1.2.…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factsupporting

confidence: 59%

“…

□

Proposition Let

A

be an EW tournament matrix of order

4 t + 1

and let

b_{1}, \dots, b_{4 t + 1}

be the invariant factors of

A + I

. Then,

b_{2 t + 1} = 1

.Proof This result follows from the argument in , Proof of Theorem 1.2 by replacing the role of

b_{2 t}

with

b_{2 t + 1}

. For the sake of the reader, we provide a proof.…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factmentioning

confidence: 89%

“…In this section, we provide proof of part (i) of Lemma 1.4. For a prime p and an integer matrix M, let M rank ( ) p denote the rank of M over the field of integers of modulo p. The following is shown in [3]. Next, we provide a strengthening of the above lemma, which is a key lemma to determine the invariant factor s t 2 +2 .…”

Section: The T (2 + 2)-nd Invariant Factor Of Skew-symmetric Ew Matmentioning

confidence: 99%

“…For a prime

p

and an integer matrix

M

, let

{rank}_{p} (M)

denote the rank of

M

over the field of integers of modulo

p

. The following is shown in .…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factmentioning

confidence: 99%

“…The Smith normal form of skew-symmetric Hadamard matrices of order 4t is uniquely determined [13,8]. Armario [3] partially determined invariant factors of a skew-symmetric EW matrix of order 4t + 2.…”

Section: Introductionmentioning

confidence: 99%

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On the Smith normal form of a skew‐symmetric d‐optimal design of order

Greaves

Suda

2018

J of Combinatorial Designs

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We show that the Smith normal form of a skew‐symmetric D‐optimal design of order n ≡ 20.3em ( mod0.3em 4 ) is determined by its order. Furthermore, we show that the Smith normal form of such a design can be written explicitly in terms of the order n, thereby proving a recent conjecture of Armario. We apply our result to show that certain D‐optimal designs of order n ≡ 20.3em ( mod0.3em 4 ) are not equivalent to any skew‐symmetric D‐optimal design. We also provide a correction to a result in the literature on the Smith normal form of D‐optimal designs.

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“…Remark The equality

b_{2 t + 1} = 1

implies that

b_{1} = \dots = b_{2 t} = 1

. Thus it follows that

s_{2} = \dots = s_{2 t + 1} = 2

, which was already shown in , Theorem 1.2.…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factsupporting

confidence: 59%

“…

□

Proposition Let

A

be an EW tournament matrix of order

4 t + 1

and let

b_{1}, \dots, b_{4 t + 1}

be the invariant factors of

A + I

. Then,

b_{2 t + 1} = 1

.Proof This result follows from the argument in , Proof of Theorem 1.2 by replacing the role of

b_{2 t}

with

b_{2 t + 1}

. For the sake of the reader, we provide a proof.…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factmentioning

confidence: 89%

Section: The T (2 + 2)-nd Invariant Factor Of Skew-symmetric Ew Matmentioning

confidence: 99%

“…For a prime

p

and an integer matrix

M

, let

{rank}_{p} (M)

denote the rank of

M

over the field of integers of modulo

p

. The following is shown in .…”

Section: The (Bold2bold-italictgoodbreakbold+bold2)‐nd Invariant Factmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Smith normal form of a skew‐symmetric d‐optimal design of order

Greaves

Suda

2018

J of Combinatorial Designs

View full text Add to dashboard Cite

show abstract

On the Smith normal form of skew E-W matrices

Cited by 1 publication

References 11 publications

On the Smith normal form of a skew‐symmetric d‐optimal design of order

On the Smith normal form of a skew‐symmetric d‐optimal design of order

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