## Linear algebra |

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### Contents

Direct sums of subspaces | 7 |

Reduction to triangular form | 22 |

Reduction to Jordan form | 31 |

Copyright | |

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### Common terms and phrases

adjoint algebra basis ofV block diagonal characteristic polynomial companion matrix complex inner product Consequently Corollary cyclic decomposition deduce deﬁne Deﬁnition Let diagonal matrix diagonalizable dimV dimW direct sum eigenvalues of f eigenvector element elementary Jordan matrix Example Consider Example Let f is diagonalizable f relative f-annihilator f-invariant subspace ﬁeld F ﬁnd ﬁnite dimension ﬁnite-dimensional inner product ﬁnite-dimensional vector space ﬁrst given Gram-Schmidt process ground ﬁeld hence inner product space invertible invertible matrix irreducible irreducible polynomials Jordan block Jordan form Jordan matrix Ker f Ker(f let f linear mapping mapping f matrix of f minimum polynomial nilpotent non-zero ﬁnite-dimensional real non-zero ﬁnite-dimensional vector ordered basis ordered orthonormal basis orthonormal subset polynomial of f positive deﬁnite projection Proof Let quadratic form readily seen real inner product self-adjoint skew-adjoint Suppose Theorem IfV Theorem Let transition matrix unique whence