advanced linear algebra - uny
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Advanced Linear AlgebraAdvanced Linear AlgebraReferences:
[A] Howard Anton& Chris Rorres. (2000). Elementary LinearAlgebra. New York: John Wiley & Sons, Inc
[B] Setya Budi, W. 1995. Aljabar Linear. Jakarta: Gramedia
R. RosnawatiJurusan Pendidikan MatematikaFMIPA UNY
.
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Inner Product SpacesInner Product Spaces
1 Inner Products 2 Angle and Orthogonality in Inner
Product Spaces 3 Gram-Schmidt Process;
QR-Decomposition 4 Best Approximation; Least Squares 5 Least Squares Fitting to Data 6 Function Approximation; Fourier
Series
1 Inner Products 2 Angle and Orthogonality in Inner
Product Spaces 3 Gram-Schmidt Process;
QR-Decomposition 4 Best Approximation; Least Squares 5 Least Squares Fitting to Data 6 Function Approximation; Fourier
Series
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1 Inner Products1 Inner Products
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2. Algebraic Properties of Inner Products2. Algebraic Properties of Inner Products
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θθ: the angle between u and v: the angle between u and v
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3 Gram3 Gram--Schmidt Process; QRSchmidt Process; QR--DecompositionDecomposition
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QRQR--DecompositionDecomposition
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4. Best Approximation; Least4. Best Approximation; LeastSquaresSquares
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Least squares solutions to ALeast squares solutions to Axx ==bb
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5 Least Squares Fitting to Data5 Least Squares Fitting to Data
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The Least Squares SolutionThe Least Squares Solution
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6 Function Approximation;6 Function Approximation;Fourier SeriesFourier Series
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Fourier Coefficients & SeriesFourier Coefficients & Series
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Fourier Approximation to y = xFourier Approximation to y = x
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Diagonalization & QuadraticDiagonalization & QuadraticFormsForms
1 Orthogonal Matrices 2 Orthogonal Diagonalization 3 Quadratic Forms 4 Optimization Using Quadratic
Forms 5 Hermitian, Unitary, and Normal
Matrices
1 Orthogonal Matrices 2 Orthogonal Diagonalization 3 Quadratic Forms 4 Optimization Using Quadratic
Forms 5 Hermitian, Unitary, and Normal
Matrices
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1 Orthogonal Matrices1 Orthogonal Matrices
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Orthonormal BasisOrthonormal Basis
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Orthogonal DiagonalizationOrthogonal Diagonalization
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Symmetric MatricesSymmetric Matrices
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Schur’s TheoremSchur’s Theorem
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Hessenberg’s TheoremHessenberg’s Theorem
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3 Quadratic Forms3 Quadratic Forms
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Conic SectionsConic Sections
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Central conics in standardCentral conics in standardpositionposition
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Definite quadratic formsDefinite quadratic forms
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Ellipse? Hyperbola? Neither?Ellipse? Hyperbola? Neither?
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4 Optimization Using Quadratic4 Optimization Using QuadraticFormsForms
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Hessian Form of theHessian Form of thesecond derivative testsecond derivative test
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5 Hermitian, Unita and Normal5 Hermitian, Unita and NormalMatricesMatrices
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Hermitian MatricesHermitian Matrices
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Unitary MatricesUnitary Matrices
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Unitarily DiagonalizingUnitarily Diagonalizinga Hermitian Matrixa Hermitian Matrix
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I. Linear TransformationsI. Linear Transformations
General Linear Transformations Isomorphisms Compositions and Inverse
Transformations Matrices for General Linear
Transformations Similarity
General Linear Transformations Isomorphisms Compositions and Inverse
Transformations Matrices for General Linear
Transformations Similarity
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General Linear TransformationsGeneral Linear Transformations
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Dilation and ContractionDilation and ContractionOperatorsOperators
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Image, Kernel and RangeImage, Kernel and Range
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Rank, Nullity and DimensionRank, Nullity and Dimension
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IsomorphismIsomorphism
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IsomorphismIsomorphism
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Compositions and InverseCompositions and InverseTransformationsTransformations
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InversesInverses
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Matrices for GeneralMatrices for GeneralLinear TransformationsLinear Transformations
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Matrix of CompositionsMatrix of Compositionsand Inverse Transformationsand Inverse Transformations
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SimilaritySimilarity
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Eigenvalues and EigenvectorsEigenvalues and EigenvectorsDefinition 1: A nonzero vector x is an eigenvector (or characteristic vector)of a square matrix A if there exists a scalar λ such that Ax = λx. Then λ is aneigenvalue (or characteristic value) of A.
Note: The zero vector can not be an eigenvector even though A0 = λ0. But λ= 0 can be an eigenvalue.
Example: The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
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Geometric interpretation ofGeometric interpretation ofEigenvalues and EigenvectorsEigenvalues and Eigenvectors
An n×n matrix A multiplied by n×1 vector x results in anothern×1 vector y=Ax. Thus A can be considered as atransformation matrix.
In general, a matrix acts on a vector by changing both itsmagnitude and its direction. However, a matrix may act oncertain vectors by changing only their magnitude, and leavingtheir direction unchanged (or possibly reversing it). Thesevectors are the eigenvectors of the matrix.
A matrix acts on an eigenvector by multiplying its magnitude bya factor, which is positive if its direction is unchanged andnegative if its direction is reversed. This factor is the eigenvalueassociated with that eigenvector.
An n×n matrix A multiplied by n×1 vector x results in anothern×1 vector y=Ax. Thus A can be considered as atransformation matrix.
In general, a matrix acts on a vector by changing both itsmagnitude and its direction. However, a matrix may act oncertain vectors by changing only their magnitude, and leavingtheir direction unchanged (or possibly reversing it). Thesevectors are the eigenvectors of the matrix.
A matrix acts on an eigenvector by multiplying its magnitude bya factor, which is positive if its direction is unchanged andnegative if its direction is reversed. This factor is the eigenvalueassociated with that eigenvector.
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6.2 Eigenvalues6.2 EigenvaluesLet x be an eigenvector of the matrix A. Then there must exist aneigenvalue λ such that Ax = λx or, equivalently,
Ax - λx = 0 or
(A – λI)x = 0
If we define a new matrix B = A – λI, then
Bx = 0
If B has an inverse then x = B-10 = 0. But an eigenvector cannotbe zero.
Thus, it follows that x will be an eigenvector of A if and only if Bdoes not have an inverse, or equivalently det(B)=0, or
det(A – λI) = 0
This is called the characteristic equation of A. Its rootsdetermine the eigenvalues of A.
Let x be an eigenvector of the matrix A. Then there must exist aneigenvalue λ such that Ax = λx or, equivalently,
Ax - λx = 0 or
(A – λI)x = 0
If we define a new matrix B = A – λI, then
Bx = 0
If B has an inverse then x = B-10 = 0. But an eigenvector cannotbe zero.
Thus, it follows that x will be an eigenvector of A if and only if Bdoes not have an inverse, or equivalently det(B)=0, or
det(A – λI) = 0
This is called the characteristic equation of A. Its rootsdetermine the eigenvalues of A.
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Example 1: Find the eigenvalues of
two eigenvalues: 1, 2Note: The roots of the characteristic equation can be repeated. That is, λ1 = λ2
=…= λk. If that happens, the eigenvalue is said to be of multiplicity k.Example 2: Find the eigenvalues of
λ = 2 is an eigenvector of multiplicity 3.
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6.2 Eigenvalues: examples6.2 Eigenvalues: examplesExample 1: Find the eigenvalues of
two eigenvalues: 1, 2Note: The roots of the characteristic equation can be repeated. That is, λ1 = λ2
=…= λk. If that happens, the eigenvalue is said to be of multiplicity k.Example 2: Find the eigenvalues of
λ = 2 is an eigenvector of multiplicity 3.
200
020
012
A
0)2(
200
020
0123
AI
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Example 1 (cont.):
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0,1
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2
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6.3 Eigenvectors6.3 EigenvectorsTo each distinct eigenvalue of a matrix A there will correspond at least oneeigenvector which can be found by solving the appropriate set of homogenousequations. If λi is an eigenvalue then the corresponding eigenvector xi is thesolution of (A – λiI)xi = 0
0,1
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Example 2 (cont.): Find the eigenvectors of
Recall that λ = 2 is an eigenvector of multiplicity 3.Solve the homogeneous linear system represented by
Let . The eigenvectors of = 2 are of theform
s and t not both zero.
0
0
0
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000
010
)2(
3
2
1
x
x
x
AI x
6.3 Eigenvectors6.3 Eigenvectors
200
020
012
AExample 2 (cont.): Find the eigenvectors of
Recall that λ = 2 is an eigenvector of multiplicity 3.Solve the homogeneous linear system represented by
Let . The eigenvectors of = 2 are of theform
s and t not both zero.
0
0
0
000
000
010
)2(
3
2
1
x
x
x
AI x
txsx 31 ,
,
1
0
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2
1
ts
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x
x
x
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6.4 Properties of Eigenvalues and Eigenvectors6.4 Properties of Eigenvalues and EigenvectorsDefinition: The trace of a matrix A, designated by tr(A), is the sumof the elements on the main diagonal.
Property 1: The sum of the eigenvalues of a matrix equals thetrace of the matrix.
Property 2: A matrix is singular if and only if it has a zeroeigenvalue.
Property 3: The eigenvalues of an upper (or lower) triangularmatrix are the elements on the main diagonal.
Property 4: If λ is an eigenvalue of A and A is invertible, then 1/λis an eigenvalue of matrix A-1.
Definition: The trace of a matrix A, designated by tr(A), is the sumof the elements on the main diagonal.
Property 1: The sum of the eigenvalues of a matrix equals thetrace of the matrix.
Property 2: A matrix is singular if and only if it has a zeroeigenvalue.
Property 3: The eigenvalues of an upper (or lower) triangularmatrix are the elements on the main diagonal.
Property 4: If λ is an eigenvalue of A and A is invertible, then 1/λis an eigenvalue of matrix A-1.
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6.4 Properties of Eigenvalues and6.4 Properties of Eigenvalues andEigenvectorsEigenvectors
Property 5: If λ is an eigenvalue of A then kλ is an eigenvalue ofkA where k is any arbitrary scalar.
Property 6: If λ is an eigenvalue of A then λk is an eigenvalue ofAk for any positive integer k.
Property 8: If λ is an eigenvalue of A then λ is an eigenvalue of AT.
Property 9: The product of the eigenvalues (counting multiplicity)of a matrix equals the determinant of the matrix.
Property 5: If λ is an eigenvalue of A then kλ is an eigenvalue ofkA where k is any arbitrary scalar.
Property 6: If λ is an eigenvalue of A then λk is an eigenvalue ofAk for any positive integer k.
Property 8: If λ is an eigenvalue of A then λ is an eigenvalue of AT.
Property 9: The product of the eigenvalues (counting multiplicity)of a matrix equals the determinant of the matrix.
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6.5 Linearly independent eigenvectors6.5 Linearly independent eigenvectorsTheorem: Eigenvectors corresponding to distinct (that is, different)eigenvalues are linearly independent.
Theorem: If λ is an eigenvalue of multiplicity k of an n n matrixA then the number of linearly independent eigenvectors of Aassociated with λ is given by m = n - r(A- λI). Furthermore, 1 ≤ m≤ k.
Example 2 (cont.): The eigenvectors of = 2 are of the form
s and t not both zero.
= 2 has two linearly independent eigenvectors
Theorem: Eigenvectors corresponding to distinct (that is, different)eigenvalues are linearly independent.
Theorem: If λ is an eigenvalue of multiplicity k of an n n matrixA then the number of linearly independent eigenvectors of Aassociated with λ is given by m = n - r(A- λI). Furthermore, 1 ≤ m≤ k.
Example 2 (cont.): The eigenvectors of = 2 are of the form
s and t not both zero.
= 2 has two linearly independent eigenvectors
,
1
0
0
0
0
1
0
3
2
1
ts
t
s
x
x
x
x