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Definitions |
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||v||
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length of v, defined by (v.v)^.5
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dimNulA
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number of free variables of A
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geometric multiplicity
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the dimension of the eigenspace corresponding to L, ie, dimNul(A - LI)
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diagonalizable
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a matrix is diagonalizable if A is similar to a diagonal matrix, ie if A = PDP-1 where P is invertible and D is a diagonal matrix
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Chapter 5 Theorem 2: Eigenvectors corresponding to distinct eigenvalues...
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are linearly independent
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subspace
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a subset of a vector space that:a. has the zero vectorb. is closed under vector additionc. is closed under scalar multiplication
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rank
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the dimension of the column space of A
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vector space
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a non-empty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to ten axioms.
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null space
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the set of all solutions to the homogeneous equation Ax = 0
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Algebraic multiplicity
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the number of times Li occurs as a root of the characteristic equation
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similar
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A is similar to B if there is an invertible matrix P such that P(-1)AP = B or, equivalently, A = PBP(-1)
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Process for QR factorization
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A = QRfind orthogonal basis for ColA = Span{x1, x2, x3}normalize vectorsQtA = R
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Chapter 6 Theorem 7: properties of orthonormal matrices
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||Ux|| = ||x||(Ux).(Uy) = x.y(Ux).(Uy) = 0 iff x.y = 0
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Orthogonal set
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A set of vectors {u1, u2, ... , up} in Rn is called an orthogonal set if ui.uj = 0 whenever i != j
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Orthogonal projection of y onto u
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Y = (y.u)/(u.u) x uY is y-hat
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What is the characteristic equation and polynomial?
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equation: det(A - LI) = 0polynomial: det(A - LI)
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The null space of an mxn matris is a subspace of...
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Rn.(the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn)
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