Linear Algebras

Cover Linear Algebras
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Hence the variables z t of character (a«) enter the same rows of the determinant S x . Hence S x is homogeneous and of degree n n or w al + 1 in the s t of character ( 1 or a = 1.
Similarly S L is homogeneous and of degree n al or n al + 1 in the variables of character («a), according as a> 1 or a = 1.
Let 3 ( being d f .
Then d x + . . . + d t = h, if 8 («>) = defines o> as an h- valued function of the x's and y's. For y 1 = 0, . . ., y k = 0, we saw that S (a>) reduces to the produc
...t (74) of its diagonal elements. Thus P t reduces to a product of d t factors xj — , ..., x h — u in some order. Hence no one appears twice in the same P t or appears in different P's. We may thus set (P 1 ) v = = (x 1 -m) ...(x p -u>) =Xi ...X p ' (76).
The factor S x of 8 is homogeneous in the variables of character (a«).


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