math A Hadamard matrix is a square matrix H of order n with entries 1 or -1 such

Question

math

A Hadamard matrix is a square matrix H of order n with entries 1 or -1 such that HH^T = nI_n, where I_n is the identity matrix and H^T is the transpose of H. If H is a Hadamard matrix of order 4t whose first row and first column contain only ones, then show that one can obtain a 2 - (4t - 1, 2t - 1, t - 1) Hadamard design from it by deleting its first row and first column. Show that if there is a 2 - (4t - 1, 2t - 1, t - 1) Hadamard design, then A_2 (4t - 1, 2t - 1) greaterthanorequalto 8t.

Explanation / Answer

A Hadamard matrix is a square matrix H of order n with entries 1 or -1 such that HHT=nIn

where In is identity matrix of order n

If H is hadamard matrix of order 4t whose first row and first column contains only ones

Claim:- To obtained Hadamard 2-(4t-1,2t-1,t-1) design

-->

Let H is hadamard matrix of order 4t .first ormalize the matrix H

step 1)

Remove the first row and first column

the 4t-1X4t-1 matrix remain .

let A has 2t-1 in each row and column

and 2t-1+1's in each row and column

so thed row and column sum's are always 1 nad -1 for A

The inner product of two distinct rows of A will be -1 & product of row with itself will be 4t-1

MAtrix equation is AJ=JAT=-J AND AAT=4tI-1 ---------------------------(1)

where I is identity matrix and J is all one matrix of the appropriate order

step 2)

Now construct matrix B=1/2 (A+J)

B is (0,1) matrix whose row and column sums are 2t-1

i.e BJ=JB=(2t-1)J

therfore matrix equation is

BBT=tI+(t-1)J is satisfies --------------------------(2)

comparing 1) and 2)

we see that B is incident matrix of symmetric

therfore there is Hadamard 2-(4t-1,2t-1,t-1) design

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.