Transcription of Linear Algebra Review and Reference
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Linear Algebra Review and Reference
Linear Algebra Review and ReferenceZico Kolter (updated by Chuong Do)October 7, 2008 Contents1 Basic Concepts and Basic Notation .. 22 Matrix Vector-Vector Products .. Matrix-Vector Products .. Matrix-Matrix Products .. 53 Operations and The Identity Matrix and Diagonal Matrices .. The Transpose .. Symmetric Matrices .. The Trace .. Norms .. Linear Independence and Rank .. The Inverse .. Orthogonal Matrices .. Range and Nullspace of a Matrix .. The Determinant .. Quadratic Forms and Positive Semidefinite Matrices .. Eigenvalues and Eigenvectors .. Eigenvalues and Eigenvectors of Symmetric Matrices .. 194 Matrix The Gradient .. The Hessian .. Gradients and Hessians of Quadratic and Linear Functions .. Least Squares .. Gradients of the Determinant .. Eigenvalues as Optimization .. 2611 Basic Concepts and NotationLinear Algebra provides a way of compactly representing and operating on sets of linearequations.
2.1 Vector-Vector Products Given two vectors x,y ∈ Rn, the quantity xTy, sometimes called the inner product or dot product of the vectors, is a real number given by xTy ∈ R = x1 x2 ··· xn y1 x2 yn Xn i=1 xiyi. Observe that inner products are really just special case of matrix multiplication.
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