{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The pseudo-inverse of a matrix A, denoted " }, { "__type": "InlineMath", "__tag": 4057, "value": "A^+" }, { "__type": "Text", "__tag": 4046, "value": ", is defined as: \"the matrix that 'solves' [the least-squares problem] " }, { "__type": "InlineMath", "__tag": 4057, "value": "Ax = b" }, { "__type": "Text", "__tag": 4046, "value": ",\" i.e., if " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\bar{x}" }, { "__type": "Text", "__tag": 4046, "value": " is said solution, then " }, { "__type": "InlineMath", "__tag": 4057, "value": "A^+" }, { "__type": "Text", "__tag": 4046, "value": " is that matrix such that " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\bar{x} = A^+b" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "It can be shown that if " }, { "__type": "InlineMath", "__tag": 4057, "value": "Q_1 \\Sigma Q_2^T = A" }, { "__type": "Text", "__tag": 4046, "value": " is the singular value decomposition of A, then " }, { "__type": "InlineMath", "__tag": 4057, "value": "A^+ = Q_2 \\Sigma^+ Q_1^T" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineMath", "__tag": 4057, "value": "Q_{1,2}" }, { "__type": "Text", "__tag": 4046, "value": " are orthogonal matrices, " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Sigma" }, { "__type": "Text", "__tag": 4046, "value": " is a diagonal matrix consisting of A's so-called singular values, (followed, typically, by zeros), and then " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Sigma^+" }, { "__type": "Text", "__tag": 4046, "value": " is simply the diagonal matrix consisting of the reciprocals of A's singular values (again, followed by zeros). 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