Applied Linear Algebra I Review page 1

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1 Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties a. Det I = 1 b. Multilinearity c. Interaction with permutations of rows and columns B. Computations with determinants 1. Laplace Theorem a. Statement and use 2. Cauchy-Binet Theorem a. Statement and use 3. Cofactors a. Rows and Columns b. Gaussian Operations to clear out a row or column 4. Adjugate Marix 5. Determinant of triangular matrices 6. Determinant of block matrices 7 Det(A*A)! 0 for A m n 8. Vandermonde determinant 9. Determinant for circulant matrices 10. Determinant of a circulant matrix 11. Determinant of a tridiagonal matrix C. Applications 1. Inverse using adjugate a. Det A " 0 if and only if A is invertible 2. Crammer s Rule 3. Cauchy-Schwarz Inequality 4. Chebyshev Polynomials 5. Characteristic Polynomials II. Vector Spaces A. Axioms 1. Two different kinds of 0 B. Examples Matrices Applied Linear Algebra I Review page 1

2 III. IV. 3. Polynomials 4. Range of a matrix 5. Kernel of a matrix C. Linear Combinations Dimensions 1. Linear Independence and Dependence 2. Span 3. Basis a. Maximal Independent Set b. Computation to find maximal independent set a. Finding the span of a set of vectors a. Unique Representation in terms of a basis b. Dimension A. Dimension of a subspace 1. Computing using Gaussian elimination a. Columns b. Rows B. Finding the Basis of a Subspace from a Spanning Set C. Kernel 1. Gaussian elimination on a matrix 1. Kernel of a matrix a. Gaussian elimination i. Read off solution with alternating 1 and rest 0 s in nonpivot positions D. Completing a linearly independent set to get a basis 1. Kernel A* or A T (real vector spaces) E. Rank of a matrix 1. Dimension of row space a. Finding a basis of the row space by Gaussian elimination 2. Dimension of Columns space b. Finding a basis of the column space by relative position of pivots 3. A = P 1 I r Q 1 for unique r a. Equivalent matrices 4. Column space of a matrix = Range of Matrix 5. rank(ab) # min(rank A, rank B) 6. rank(a + B) # rank A + rank B F. Null Space of a matrix 1. Finding the null space 2. rank A + nullity A = number of columns of A Solutions of lineear equations A. General solution of a set of linear equations 1. Particular solution + Kernel Applied Linear Algebra I Review page 2

3 2. Least squares solution to overdetermined system 3. Underdetermined system a. Minimal norm to an uderdetermined system B. Gaussian elemination 1. Drive to upper triangular form C. Inverse Matrix 1. Computing Inverse V. Linear Transformation A. Definition 1. Well-defined, i.e., is a function B. Matrix of a linear Transformation C. Range D. Kernel 1. Defect E. Rules of Combination 1. Addition and Subtraction 2. Composition a. Multiplication F. Factoring out the kernel VI. Change of Basis A. Linear Transformations 1. Matrix of a linear transformation with respect to two bases a. A = [T], means T($% i v i ) =$ & i w i here A% = & b. Images in the column c.. A = [TX] = [T], [X] 2. Matrix P, = [v 1,, v n ] - 1 [w 1,, w n ] changes from to 3. $% i v i =$ & i w i if P% = & B. Change from [A], to [A] ', ' 1. [A] ', ' = P, ' [A], ( ', C. Similarity 1. A ~ B if and only if A and B represent the same linear transformation with respect to different bases 1. Represent the same linear transformation with respect to different bases D. Changing bases of a subspace VII. Orthogonalization A. Inner product spaces 1. Complex value inner product a. Hermitian form 2. Real Inner product 3. Parallelogram law B. Orthogonal Vectors Applied Linear Algebra I Review page 3

4 VIII 1. Gram Schmidt Process C. Orthogonal Complement 1. Finding S ) 2. V+W a. (S + T) ) = S ) *T ) b. (S*T) ) = S ) + T ) c. S )) = [S] 2. Orthogonal complement of Range of matrix A = column space A is kernel A* D. Projection on a subspace 1. proj S x = $<x, q i > q i where {q i } is an orthonormal basis of S 2. Matrix of projection is [q 1,, q m ]q 1,, q m ] * where {q i } is an orthonormal basis of S 3. Closest point in subspace to given point = geometric projection E. Householder Matrices 1. Forming Householder Matrices 2. Using Householder Matrices to reduce to upper Hessenberg form 3. u,v F. QR factorization a. U w with 0 s in the first entries Eigenvalues and Eigenvectors A. Definition of Eigienvalue A is not invertible 2. Spectrum 3. Root of the characteristic Equation B. Eigenvalues for special kinds of matrices 1. Selfadjoint (real eigenvalues) 2. Diagonal 3. Upper triangular 4. Circulant C. Finding the Characteristic equation 1. Coefficients in the characteristic equation a. Trace = a n - 1 b. Determinant = a 0 c. a k = E k (A) = S k (- 1,, - n ) where E k (A) = sum of the n principal k k minors and k S k (- 1,, - n ) = $ 1 # i1 < < i k # n.- i - k ik for /(A) = {- 1,, - n }. 2. Relationship between the characteristic equations of AB and BA C. Invariance of Characteristic Equation and Eigenvalues 1. Similarity transform 2. Trace (A k ) = Trace (B k ) a. Newton's formula for coefficients of polynomial in terms of symmetric functions Applied Linear Algebra I Review page 4

5 IX XI. 3. Leverierr s Algorithm for characteristic polynomial c n - n + c n n c 0 a. c n = 1 and C n - 1 = I b. c n - j = - (1/j) tr (AC n - j ) and C n - j - 1 = C n - j A + c n - j I 4. Spectral Mapping theorem 1. /(A - 1 ) = /(A) - 1 D. Multiplicity of eigenvalues 1. Algebraic and Geometric matrix a. Nonderogatory matrix b. Simple Matrix c. Diagonalizable matrix 2. Lemma in proof of geometric multplicity # algebraic multiplicity concerning eigenvalues of submatrices E. Diagonalizable 1. Basis of eigenvectors 2. Simple matrix 3. Nondefective matrix 4. Square root if A is invertible E. Simultaneously diagonalizable family 1. Each element in is diagonalizable and are mutually commuting 2. A is simple and A and B commute a. B commutes with a diagonalizable matrix F. Commuting family of matrices 1. Common eigenvector 2. Simultaneously diagonalizable when each matrix is diagonalizable Schur's Triangulization Theorem A. Statement 1. Triangular matrix 2. Unitary matrix 3. Simultaneous reduction of commuting families to triangular form using commaon eigenvectors B. Use in proving theorems about eigenvectors and eigenvalues C. Reduction using eigenvector 1. Deflation procedure D. Diagonalization of Hermitian matrices 1. Obtained from Schur's Triangularization Theorem E. Cayley Hamilton Theorem 1. Evaluating polynomials of matrices in terms of polynomials of lower degree a. Lagrange polynomial (simple matrices) b. Remainder from long division Normal and Unitary Matrices A. Householder matrices 1. Reduction to Hessenberg form 2. Reduction to triadiagonal 3. Reduction to triangular by left multiplication Applied Linear Algebra I Review page 5

6 IX XII. XIII. Singular Value Decomposition A. Singular values 1. Nonzero eigenvalues for A with multiplicity 2. Singular values for A = Singular values for A* B. UDV* = A 1. Computing U and V C. Uniqueness of U, V and D D. Using only large singular values for an approximation E. Applications 1. Data Compression Jordan Form A. Polynomials 1. Characteristic polynomial a. Generalized eigenvalues 2. Minimal polynomial a. Algebraic multiplicity b. Geometric multiplicity B. Size of Jordan Blocks C. Finding Basis that will bring to Jordan Form 1. Nilpotent Matrices D. Applications to Autonomous Systems of Differential Equations E. Functions of matrices 1. Every f(a) is equal to p(a) for an interpolating polynomial 2. Computation of p(a) using Jordan form Hermitian Matrices A. Types 1. Skew symmetric and skew hermitian 2. Hermitian 3. linear combination B. Matrices similar to a hermitian matrix 1. Real matrices are similar to their inverse C. Positive Definite D. Quadratic Forms 1. Principal Axis Theorem Applied Linear Algebra I Review page 6