By Maud King Murphy
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Extra info for A Beginner's Guide to the Stars
If A is any matrix with m rows and n columns and B is any non-singular, n-rowed, square matrix, then A and AB have the same rank. If C is any non-singular, m-rowed, square matrix, then A and CA have the same rank. For, if r is the rank of A and p is the rank of P = AB, the Corollary gives p ^ r and, when applied to A = PB~X, gives r ^ p, whence r = p. Next, if r' is the rank of CA, the Corollary gives r' ^ r and, when applied to A = C~x • CA, gives r ^ r', whence rf = r. 27. Bilinear forms. A polynomial in the m + n variables Xi,.
Od&i T h e o r e m 2 . If S is both homogeneous (of degree d) and isobaric (of weight w) in ao, . . , ap, then (8) (&0 — Oil)S = nS, n = pd — 2w. This result is the case r = 1 of (9) (ttOr — Or Q,)S = r(n — r + l ) O r_1 S. To prove the latter by induction on r, assume that it holds when r = k, and note that OS is of degree d and weight w + 1, so that when (9) is employed with S replaced by OS, n must be replaced b y n — 2. Hence (i20*+1 - Ok+1 Q)S = ( M k - Oh ti)OS + Ok(tiO - OQ)S = k ( n - 2 - k + 1 )0 * “ 1 OS + nOkS = (k + l ) ( n - k ) O kS, and (9) holds for r = k + 1.
That with i = s + n — 1 by qi, that with i = s + n by unity, and add. The coefficient of pk is ric9 qn + rk*+l qn_i + ------ b rk*+n~l qx + rks+n, which is the product of rk8 by the zero value of the polynomial (19) for z = rk. The new left member is evidently (18). The equation having the roots rh . . ,rn is evidently obtained by eliminating qu . . , qn from (18) and (19). We prefer the § 37 CANONICAL FORM OF ODD ORDER 20] homogeneous equation in x and y whose linear factors are x + y (i = 1, .
A Beginner's Guide to the Stars by Maud King Murphy