矩阵笔记 from Linear Algebra and Its Applications and more

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4.1. What makes rank, determinant and eigenvalues unchanged ........................................... 5
4.1.1. Row reduction .......................................................................................................... 5
4.1.2. CTAC ....................................................................................................................... 5
4.1.3. Pivots, diagonal entries and eigenvalues .................................................................. 6
4.1.4. (A-1)T = (AT)-1 ....................................................................................................... 7
4.2. Four subspaces ................................................................................................................ 7
4.2.1. Row Reduction will change the column space ...................................................... 10
4.2.2. Orthogonal decomposition induced by A ................................................................ 9
4.2.3. Ax=b is solvable iff yTb=0 whenever yTA=0 ....................................................... 11
4.2.4. Projection matrix .................................................................................................... 13
4.3. Rank .............................................................................................................................. 15
4.3.1. rank(AB)<= min{ rank(A), rank(B) } .................................................................... 15
4.3.2. Left/Right-multiplication of an invertible matrix keeps rank unchanged .............. 16
4.3.3. Full rank ................................................................................................................. 17
4.3.4. dim(column space) = dim(row space) = rank(A) ................................................... 17
4.3.5. rank(AAT)=rank(ATA)=rank(A) .......................................................................... 17
4.3.6. Left-inverse and right-inverse ................................................................................ 19
4.3.7. rank(CTAC) = rank(A) .......................................................................................... 20
4.3.8. Effective Rank........................................................................................................ 19
4.3.9. rank(A)=number of non-zero singular values ........................................................ 16
4.3.10. Relation to Ax=b ............................................................................................. 20
4.3.11. Rank1 matrix ................................................................................................... 14
4.3.12. rank(X+Y)>= min{rank(X), rank(Y)} ............................................................ 16
4.4. Trace............................................................................................................................. 21
4.4.1. tr(A)=sum(eigenvalues) ......................................................................................... 21
4.4.2. tr(AB)=tr(BA) ........................................................................................................ 22
4.4.3. tr(A+B)=tr(A)+tr(B) .............................................................................................. 23
4.4.4. tr(alphaA)= alpha tr(A) .......................................................................................... 23
4.4.5. tr(AT)= tr(A) when A is square ............................................................................. 24
4.4.6. tr(vvT)= ||v||^2 ........................................................................................................ 24
4.4.7. xTy = tr(yxT) ......................................................................................................... 24
4.4.8. xTAy = tr(A yxT) ................................................................................................... 24
4.4.9. Inner product of matrices ....................................................................................... 25
4.5. Determinant ................................................................................................................... 26
4.5.1. Three axioms .......................................................................................................... 26
4.5.2. det(A)=product(eigenvalues) ................................................................................. 27
4.5.3. At least one eigenvalues is zero if A is singular .................................................... 28
4.5.4. Product of pivots .................................................................................................... 2
4.5.8. det(AT) = det A ...................................................................................................... 29
4.5.9. det(I+alpha vvT) = 1+alpha ................................................................................... 29
4.5.10. det(I+uvT) = 1+uTv ........................................................................................ 29
4.5.11. det(I+epsilon A) ~= 1+epsilon tr(A) ............................................................... 30
4.6. Eigenvalues ................................................................................................................... 31
4.6.1. Row reduction changes eigenvalues ...................................................................... 32
4.6.2. Eigenvalues(AT) = Eigenvalues(A) ....................................................................... 32
4.6.3. Largest and smallest eigenvalues ........................................................................... 33
4.6.4. Eigenvalues(AB)~=Eigenvalues(BA) .................................................................... 35
4.6.5. Distinct Non-zero eigenvalues=>linear independent eigenvectors ........................ 37
4.6.6. Similar matrices share the same eigenvalues ......................................................... 37
4.6.7. Largest and smallest singular values ...................................................................... 38
4.6.8. Condition number .................................................................................................. 38
4.6.9. A<=B iff lambdaA<=lambdaB .............................................................................. 38
4.7. Definiteness ................................................................................................................... 39
4.7.1. Definiteness of a block matrix ............................................................................... 41
4.7.2. Positive semidefiniteness on Restricted set ........................................................... 41
4.8. Hermitian matrices ........................................................................................................ 42
4.8.1. Generalized symmetric matrices ............................................................................ 42
4.8.2. xH A x is real ......................................................................................................... 43
4.8.3. All eigenvalues are real .......................................................................................... 43
4.8.4. Eigenvectors from different eigenvalues are orthogonal ....................................... 43
4.8.5. All eigenvectors are orthogonal ............................................................................. 43
4.9. Real symmetric matrices ............................................................................................... 44
4.8.1. Real symmetric matrices are Hermitian ................................................................. 44
4.8.2. All eigenvectors are real ........................................................................................ 44
4.8.3. Q ^ QT.................................................................................................................... 44
4.10. Real matrices ................................................................................................................. 44
4.9.1. Eigenvalues can be complex .................................................................................. 44
4.9.2. Eigenvectors from real eigenvalues are real .......................................................... 44
4.9.3. Complex eigenvectors are conjugate pairs............................................................. 45
4.11. Unitary matrices ............................................................................................................ 45
4.10.1. Definition of orthogonal matrix ...................................................................... 45
4.10.2. Generalized orthonormal matrices .................................................................. 46
4.10.3. Length unchanged ........................................................................................... 46
4.10.4. |eigenvalues|=1 ................................................................................................ 46
4.10.5. Eigenvectors from different eigenvalues are orthonormal .............................. 46
4.12. Triangular Forms by Unitary ........................................................................................ 46
4.11.1. U-1 A U is diagonal iff eigenvectors of A are orthogonal .............................. 47
4.13. Dec: Symmetric semidefinite squareroot ...................................................................... 48
4.14. Dec: rank1 matrix multiplication .................................................................................. 48
4.15. Dec: Diagonalizability .................................................................................................. 49
4.13.1. The SN condition of diagonalizable................................................................ 49
4.16. Dec: Spectral Decomposition for Hermitian matrix ..................................................... 48
4.17. Dec: Generalized eigenvalue decomposition ................................................................ 52
4.18. Dec: SVD ...................................................................................................................... 54
4.16.1. Definitions ....................................................................................................... 54
4.16.2. Left and right singular vectors ........................................................................ 54
4.16.3. Largest and smallest singular values ............................................................... 55
4.16.4. AAT and AT A are semidefinite ..................................................................... 56
4.16.5. 4 fundamental subspaces ................................................................................. 56
4.16.6. Pseudo-inverse ................................................................................................ 59
4.19. Dec: QR decomposition ................................................................................................ 61
4.17.1. Square matrix .................................................................................................. 62
4.17.2. Rectangular matrix .......................................................................................... 62
4.20. Dec: LU decomposition ................................................................................................ 63
4.18.1. PA=LDR ......................................................................................................... 63
4.18.2. LU decomposition with only one column vector changed ............................. 64
4.21. Dec: Pivot decomposition ............................................................................................. 64
4.22. Jordan form ................................................................................................................... 65
4.20.1. Block diagonal square matrix ......................................................................... 65
4.20.2. Size of a block ................................................................................................. 65
4.20.3. Jordan block and Jordan form ......................................................................... 65
4.20.6. Jordan form by Strang ..................................................................................... 69
4.23. Block matrix .................................................................................................................. 70
4.21.4. Inverse of block matrix ................................................................................... 74
4.21.5. LU decomposition ........................................................................................... 76
4.24. Schur complement......................................................................................................... 77
4.22.1. Definition of Schur complement ..................................................................... 77
4.22.2. Minimization of quadratic form ...................................................................... 77
4.22.3. Definiteness of a block matrix ........................................................................ 78
4.25. Vibration Methods ........................................................................................................ 82
4.23.1. Solve singular equation ................................................................................... 82
4.23.2. Matrix inverse with only one column vector changed .................................... 84
4.26. Least squares ................................................................................................................. 85
4.24.1. When A has dependent columns ..................................................................... 85
4.24.2. Examples of shortest length approximate solutions ........................................ 86
4.24.3. Shortest solution of ATAx=ATb is always in rowspace of A ........................ 87
4.24.4. Formula of pseudoinverse ............................................................................... 88

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