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.”