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2. The Symmetric Product of Matrices Here an: seme denitiens and results related te the new pro-duet intreduwd in [2]. For a positive integer 1: let

Could some one help me to use definition 1 to find the symmetric product AⵙB of the matrices A and B in example 1 in details ( I do not understand how they used the definition her!) IMG_1308.jpg

IMG_1308.jpg

2. The Symmetric Product of Matrices Here an: seme definitiens and results related te the new pro-duet intreduwd in
[2]. For a positive integer 1: let I“ stand fer all ruw a-tuples with nennega—
the integer entries with the fellewing linear order: 3 = {31, fig. fin) <2 3 =
(01.02,...1nn] if and enly if Ifil <2 lal 0r lfil = |nr| and fil :h r1101“ lfil = led.
31 = in and ,3: 3.:- nr: eteetera1 where |nr| stands for en + a: + + at”. It is clear that for £1,131”? E In one has :1 {i ,3 if and only ifrx + 'T *5 )3 + ’T.
when: the sum means the wmpenent-wise sum. We write ,6 e: er if ,6; 1: n; for alli: 1, 2......11, ('1 ,tilfuwfiH" )3 ) stands for . “l . a! = alleginnfli. A RADIUS 0F ABSOLUTE CUNVERG ENCE... 4E5 In future. n. and n’ are assuunsi be be any final pesitiye integers. Let F
stand for the field of real er template: numbers. Fer any nennegatiye inntger numbers :1 19’ let M“: “(If p: F) = M (mp: F}
stand let all "p" x :1" size matrices A = (A? llel—P |n'|—11' (e’ presents new1 a»
presents eelumn and e E Ime’ E In!) with entries l'rem F I
. . . . 1 — 1 1 — l The erdlnal‘y srze ei'sueh matrix Is ( if I, i 1 ) x ( l1: : 1 ). fiver
sueh kind umtrites in additien to the ordinary sum and product of matrices we
wnsider the fellewing ”preduet" G) as well: Definition 1. If A E MUJ’JJ:F) and B E J'l'fldfifiF} then AGE =
['3' E M (p’ + q’. p + 13:17} sueh that i'er any |nr| = p + q. le’l = p’ + q’. where e E Ime’ E fur.
u' _ If :1 '_—fi'
(3‘” _ Z (" a )Afi e
we . where the. sum is taken ewtr all J? E In. If E fur. for which lfil = p. |fi’| =
fl :3: e and 6’ *3: rr’. Example 1. If n. = n.’ = 2. p = 4"; =1 in erdinary netatiens
i'er matriees A = ( a” a.” )1 B— - (l h”: (Ill: ) the preduet AGE can he
“2| “2: b2. be:
giwtn as
«40:3 =
2"will!“ “Ill-'12 + fuel?“ 2%th: Elflunl'nl + Halli”) fl-Illlez + Heel!” +fl|2l12|+ Hell”: Elf-Hem: + (1225”)
2"Jelllel “21 be: + #22521 gunk.”

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