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THE THEORY OF SIMULTANEOUS
ORTHOGONAL PROJECTIONS AND LEAST SQUARES
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algorithm apply axis back-substitution basis choose coefficient coeﬂicient column space combination completely components compute condition number constraints corner corresponding cost deﬁned determinant differential equation dimension eigen eigenvalues eigenvectors equal error exactly example EXERCISE factor feasible set ﬁnal ﬁnally ﬁnd ﬁnite ﬁrst column ﬁrst equation ﬁrst row formula free variables Gaussian elimination gives Gram—Schmidt Hermitian identity matrix inequality inﬁnitely inner product inverse Jordan form least squares length linear algebra linear programming linearly independent lower triangular main diagonal matrix multiplication minimize minimum nonnegative nonsingular nonzero nullspace optimal original orthogonal matrix orthonormal permutation permutation matrix perpendicular pivots plane polynomial positive deﬁnite problem projection proof properties pseudoinverse rank row exchanges row space satisﬁes simplex method singular solution solve spanned strategy subspace subtracted Suppose symmetric matrices symmetric matrix system Ax theorem tion transformation transpose unitary unknowns upper triangular vector space xTAx