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Equivalence Rank and Inverses
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arbitrary called canonical matrix canonical set characteristic polynomial characteristic roots characteristic vectors coefficient matrix column rank completes the proof complex number compute congruent coordinates Corollary defined degree denote determined diag diagonal elements diagonal matrix direct sum elementary divisors elementary matrices equal Exercises ff[x fi(x field ff Fn(ff form a basis formula hence Hermitian matrices integer interchanges irreducible left inverse Lemma linear combination linear transformation linearly independent matric polynomial matrix over ff minimum polynomial multiplication nomials nonsingular matrix normal matrices null space obtained orthogonal matrices orthogonally similar orthonormal basis P-lAP P'AP pi(x polynomial f(x positive definite product of elementary quadratic form real numbers relatively prime result row h row rank scalar field Section Show similarity invariants skew matrix square matrix subdeterminants submatrix subspace symmetric matrices Theorem 3-3 transpose unique unitarily similar unitary vector space whence zero