Revision #1571 → #1800 · back to history
modifiedNon-uniqueness of the SVDc159d3ede351
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| anchors | [{"section":"Example","snippet":"This particular singular value decomposition is not unique"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {V} ^{*}={\\begin{bmatrix}\\color {Violet}0&\\color {Violet}0&\\color {Violet}-1&\\color {Violet}0&\\color {Violet}0\\\\\\color {Plum}-{\\sqrt {0.2}}&\\color {Plum}0&\\color {Plum}0&\\color {Plum}0&\\color {Plum}-{\\sqrt {0.8}}\\\\\\color {Magenta}0&\\color {Magenta}-1&\\color {Magenta}0&\\color {Magenta}0&\\color {Magenta}0\\\\\\color {Orchid}{\\sqrt {0.4}}&\\color {Orchid}0&\\color {Orchid}0&\\color {Orchid}{\\sqrt {0.5}}&\\color {Orchid}-{\\sqrt {0.1}}\\\\\\color {Purple}-{\\sqrt {0.4}}&\\color {Purple}0&\\color {Purple}0&\\color {Purple}{\\sqrt {0.5}}&\\color {Purple}{\\sqrt {0.1}}\\end{bmatrix}},}"}] | — |
modifiedCompact SVD of the example matrixb11f5664cc96
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| anchors | [{"section":"Example","snippet":"eliminates these superfluous rows, columns, and singular values"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {U} _{r}&={\\begin{bmatrix}\\color {Green}0&\\color {Blue}-1&\\color {Cyan}0\\\\\\color {Green}-1&\\color {Blue}0&\\color {Cyan}0\\\\\\color {Green}0&\\color {Blue}0&\\color {Cyan}0\\\\\\color {Green}0&\\color {Blue}0&\\color {Cyan}-1\\end{bmatrix}}\\\\[6pt]\\mathbf {\\Sigma } _{r}&={\\begin{bmatrix}3&0&0\\\\0&{\\sqrt {5}}&0\\\\0&0&2\\end{bmatrix}}\\\\[6pt]\\mathbf {V} _{r}^{*}&={\\begin{bmatrix}\\color {Violet}0&\\color {Violet}0&\\color {Violet}-1&\\color {Violet}0&\\color {Violet}0\\\\\\color {Plum}-{\\sqrt {0.2}}&\\color {Plum}0&\\color {Plum}0&\\color {Plum}0&\\color {Plum}-{\\sqrt {0.8}}\\\\\\color {Magenta}0&\\color {Magenta}-1&\\color {Magenta}0&\\color {Magenta}0&\\color {Magenta}0\\end{bmatrix}}.\\end{aligned}}}"}] | — |
modifiedSingular value characterization334a9e4aea6f
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| anchors | [{"section":"Singular values, singular vectors, and their relation to the SVD","snippet":"is a singular value for"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {Mv} &=\\sigma \\mathbf {u} ,\\\\[3mu]\\mathbf {M} ^{*}\\mathbf {u} &=\\sigma \\mathbf {v} .\\end{aligned}}}"}] | — |
modifiedDiagonal entries equal singular values697a97a965a0
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| anchors | [{"section":"Singular values, singular vectors, and their relation to the SVD","snippet":"are equal to the singular values of"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {M} =\\mathbf {U} \\mathbf {\\Sigma } \\mathbf {V} ^{*}}"}] | — |
modifiedSVD relations to eigendecomposition91f814fc1b4b
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| anchors | [{"section":"Relation to eigenvalue decomposition","snippet":"the following two relations hold"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {M} ^{*}\\mathbf {M} &=\\mathbf {V} \\mathbf {\\Sigma } ^{*}\\mathbf {U} ^{*}\\,\\mathbf {U} \\mathbf {\\Sigma } \\mathbf {V} ^{*}=\\mathbf {V} (\\mathbf {\\Sigma } ^{*}\\mathbf {\\Sigma } )\\mathbf {V} ^{*},\\\\[3mu]\\mathbf {M} \\mathbf {M} ^{*}&=\\mathbf {U} \\mathbf {\\Sigma } \\mathbf {V} ^{*}\\,\\mathbf {V} \\mathbf {\\Sigma } ^{*}\\mathbf {U} ^{*}=\\mathbf {U} (\\mathbf {\\Sigma } \\mathbf {\\Sigma } ^{*})\\mathbf {U} ^{*}.\\end{aligned}}}"}] | — |
modifiedPseudoinverse via SVDdc73b4fb03d8
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| anchors | [{"section":"Pseudoinverse","snippet":"The pseudoinverse of the matrix"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {M} ^{+}=\\mathbf {V} {\\boldsymbol {\\Sigma }}^{+}\\mathbf {U} ^{\\ast },}"}] | — |
modifiedEckart–Young theorema29c2855688f
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| anchors | [{"section":"Low-rank matrix approximation","snippet":"This is known as the Eckart–Young theorem"},{"type":"math_alttext","value":"{\\displaystyle {\\tilde {\\mathbf {M} }}=\\mathbf {U} {\\tilde {\\mathbf {\\Sigma } }}\\mathbf {V} ^{*},}"}] | — |
modifiedOrthogonal Procrustes problemae35e2e4d297
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| anchors | [{"section":"Nearest orthogonal matrix","snippet":"the orthogonal Procrustes problem"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {Q} ={\\underset {\\Omega }{\\operatorname {argmin} }}\\|\\mathbf {A} {\\boldsymbol {\\Omega }}-\\mathbf {B} \\|_{F}\\quad {\\text{subject to}}\\quad {\\boldsymbol {\\Omega }}^{\\operatorname {T} }{\\boldsymbol {\\Omega }}=\\mathbf {I} ,}"}] | — |
modifiedLargest eigenvalue as maximum of Rayleigh quotiente4ac25a1cf43
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| anchors | [{"section":"Proof of existence","snippet":"this continuous function attains a maximum"},{"type":"math_alttext","value":"{\\displaystyle \\nabla \\mathbf {u} ^{\\operatorname {T} }\\mathbf {M} \\mathbf {u} -\\lambda \\cdot \\nabla \\mathbf {u} ^{\\operatorname {T} }\\mathbf {u} =\\mathbf {0} }"},{"type":"math_alttext","value":"{\\displaystyle \\nabla \\mathbf {x} ^{\\operatorname {T} }\\mathbf {M} \\mathbf {x} -\\lambda \\cdot \\nabla \\mathbf {x} ^{\\operatorname {T} }\\mathbf {x} =2(\\mathbf {M} -\\lambda \\mathbf {I} )\\mathbf {x} .}"}] | — |
modifiedSpectral theorem yields unitary diagonalizationf5c5750d5de0
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| anchors | [{"section":"Based on the spectral theorem","snippet":"by the spectral theorem , there exists an"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {V} ^{*}\\mathbf {M} ^{*}\\mathbf {M} \\mathbf {V} ={\\bar {\\mathbf {D} }}={\\begin{bmatrix}\\mathbf {D} &0\\\\0&0\\end{bmatrix}},}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}\\mathbf {V} _{1}^{*}\\\\\\mathbf {V} _{2}^{*}\\end{bmatrix}}\\mathbf {M} ^{*}\\mathbf {M} \\,{\\begin{bmatrix}\\mathbf {V} _{1}&\\!\\!\\mathbf {V} _{2}\\end{bmatrix}}={\\begin{bmatrix}\\mathbf {V} _{1}^{*}\\mathbf {M} ^{*}\\mathbf {M} \\mathbf {V} _{1}&\\mathbf {V} _{1}^{*}\\mathbf {M} ^{*}\\mathbf {M} \\mathbf {V} _{2}\\\\\\mathbf {V} _{2}^{*}\\mathbf {M} ^{*}\\mathbf {M} \\mathbf {V} _{1}&\\mathbf {V} _{2}^{*}\\mathbf {M} ^{*}\\mathbf {M} \\mathbf {V} _{2}\\end{bmatrix}}={\\begin{bmatrix}\\mathbf {D} &0\\\\0&0\\end{bmatrix}}.}"}] | — |
modifiedOne-sided Jacobi algorithm66f9fa3a707e
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| anchors | [{"section":"One-sided Jacobi algorithm","snippet":"One-sided Jacobi algorithm is an iterative algorithm"},{"type":"math_alttext","value":"{\\displaystyle M\\leftarrow MJ(p,q,\\theta ),}"}] | — |
modifiedTwo-sided Jacobi algorithm72f56b0ffa55
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| anchors | [{"section":"Two-sided Jacobi algorithm","snippet":"is an iterative algorithm where a square matrix"},{"type":"math_alttext","value":"{\\displaystyle M\\leftarrow J^{T}GMJ}"}] | — |
addedPolar decomposition (operators)013647d78833
addedCompact self-adjoint operators are diagonalizablea5df6ab6e00e