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Householder Matrix H( W )= E ( W , W ,2) = I − 2 Wwh for Wh W =1

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Householder Matrix H( W )= E ( W , W ,2) = I − 2 Wwh for Wh W =1 Outline Elementary Matrix Full-rank Decomposition Triangular Matrix Decomposition QR Decomposition Schur Theorem and Normal Matrix 1 Elementary Matrix Elementary Matrix E(,, u vσσ )= I − uvH Identity matrix plus a rank-one matrix Properties on the Elementary Matrix Determinant Euv(,,σσ )= 1 − vH u σ inverse matrix Euv(,,σ )−1 = Euv (,, ) σ vuH −1 for any two nonzero vectors a and b, their exists an elementary matrix such that Euv(,,σ ) a= b 2 Elementary Matrix Determinant Euv(,,σσ )= 1 − vH u uH uuHH u H E(,, u v σσ )=−=−uvHH u σ v , u uu u H uuHH −=−uσσ v1, vu uu uH E(,,) u vσ = u HH if u u = 0. 0 00 σ Inverse Matrix Euv(,,σ )−1 = Euv (,, ) σ vuH −1 brute-force proof For any two nonzero vectors a and b, their exists an elementary matrix such that E(u,v,σ )a = b 3 Elementary Matrix For any two nonzero vectors a and b, their exists an elementary matrix such that Euv(,,σ ) a= b E(,, u vσσ ) a=−= a uvH a b ⇒=−σuvH a a b ⇒=−u a b,1σ vaH = 4 Elementary Matrix Householder Matrix H( w )= E ( w , w ,2) = I − 2 wwH for wH w =1 Properties on the Householder Matrix determinant det(Hw ( ))=−=− 1 2 wwH 1 inverse Hw()−1 = Hw ()H = Hw () 5 Full-Rank Decomposition Full-rank Decomposition Theorem: For m*n matrix A and det(A) = r, there exists m*r full- rank matrix B and r*n full-rank matrix C, such that A = BC. Proof: SVD of matrix A, A = UΣVH find the non-zero elements of Σ 6 Triangular Matrix Decomposition LU Decomposition: There exists a LU decomposition A = LU for a non-singular matrix A, where L is an unit lower triangular matrix and U is an upper triangular matrix, if and only if a11 a12 ... a1k a a .. a A = 21 22 2k ≠ 0 k ... ... ... ... ak1 ak 2 ... akk for all k. 7 Triangular Matrix Decomposition LDU Decomposition: There exists a LDU decomposition A = LDU for a non-singular matrix A, where L is an unit lower triangular matrix, U is an unit upper triangular matrix, and D is a diagonal matrix if and only if a11 a12 ... a1k a a .. a A = 21 22 2k ≠ 0 k ... ... ... ... for all k. ak1 ak 2 ... akk 8 Triangular Matrix Decomposition LU Decomposition: Construction Preliminary Row Transform for Matrix A, from bottom to top, converting it to a upper-triangular −1 matrix NN ∏∏LAii=⇒= U A L U ii=11= Preliminary Column Transform for Matrix U, from left to right, converting it to a diagonal matrix −−11 NN NN ∏∏LAii U=⇒= D A ∏ Li D ∏ U i ii=11 = i=1 i= 1 9 QR Decomposition QR Decomposition: For a non-singular matrix A, there exists an unitary matrix Q and an upper triangular matrix R, such that A = QR. Proof: the Gram-Schmidt orthogonalization on the columns of matrix A 10 Schur Theorem Unitary similarity: A and B are unitary similar, if and only if B = UHAU, for an unitary matrix U Schur Theorem: For any n*n matrix A, there exists an unitary matrix U and an upper- triangular matrix R, such that UHAU = R. Proof: Mathematical induction, left for homework 11 Normal Matrix Normal Matrix: AAH = AHA Matrix A is Unitary similarity to a diagonal matrix, if and only if A is a normal matrix. Proof: based on the Schur Theorem, prove that the upper triangular matrix is diagonal, left for homework 12 Normal Matrix Diagonization of Normal Matrix Matrices A and B are normal matrices, and AB = BA, then there exists an unitary matrix U such that UHAU and UHBU are both diagonal. Proof: left for homework 13.
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