By Otto Schreier, Emanuel Sperner
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Extra info for Projective Geometry of N Dimensions (Volume Two of Introduction to Modern Algebra and Matrix Theory)
Example text
Proof: Let A be left invertible. Then there is an n x n matrix B such that BA = I< Then H is cogredient to a diagonal matrix (ai \ I I' \ o Since S is nonalterate, there is a nonzero diagonal element of 5, say s u ^ 0. Interchanging the first row and the i-th row of S and the first column and the i-th column of S simultaneously, we obtain a symmetric matrix which is cogredient to S and whose element at (1, 1) position is nonzero. Therefore we can assume that s u ^ 0. Write 5 = | ( 311 u V 522 / where U u — and5*22 S22= =(St*)2