Matrix Algebra: Summary
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1 May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices E.. Notation E..2 Special Types of Vectors E..3 Special Types of Matrices E..4 Vector Products E..5 Useful Matrix Theorems E.2 Eigenvalue Problem E.2. Properties and Formulas for Eigenvalues and Eigenvectors E.2.2 Useful Eigenvalue Theorems The following is a brief summary of notation and facts from matrix algebra that are relevant to the contents of this textbook. Some of this material is in the text, and is included here for completeness. Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
2 E. Vectors and Matrtices E.. Notation Vectors: bold lower case letters v, x, etc. All vectors are assumed to be written in column format (see below) Matrices: bold upper case letters A, R, etc(seebelow) v a a 2 a n v 2 a 2 a 22 a 2n v = B A = A v n a m a m2 a mn Transpose: designated using a superscript T a a 2 a m v T = v v 2 v n A T a 2 a 22 a m2 = B a n a 2n a mn Important properties: (AB) T = B T A T,(A + B) T = A T + B T,(A T ) T = A, and (Av) T = v T A T E..2 Special Types of Vectors Independent Vectors: v, v 2,, v k are independent if it is not possible to write any one of them as a linear combination of the others. More mathematically, they are independent if the only numbers c, c 2,, c k that satisfy c v + c 2 v c k v k = are c = c 2 = = c k =. Orthogonal Vectors: two vectors v and w are orthogonal if and only if v w = Orthonormal Vectors: v, v 2,, v k are orthonormal if they are mutually orthogonal (so, v i v j =, 8i 6= j) and v i v i =, 8i Unit Vector: this is a vector with unit length, which means that v = Zero Vector: this is the vector containing only zero entries. This vector is denoted as. E..3 Special Types of Matrices Diagonal Matrix: this is a square matrix with the only nonzero entries on the diagonal d d 2 D = B d n Worth mentioning: one or more of the d i s can be zero 2 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
3 Identity Matrix: this is a diagonal matrix with d i =, 8i. This matrix is denoted as I. I = B Inverse Matrix: if A is square (and invertible), then inverse is denoted as A and it has the property that A A = I (note that it is also true that AA = I). Important properties: (AB) = B A and (A ) = A Lower Triangular Matrix: one in which all entires above the diagonal are zero Worth mentioning: if A is lower triangular then A T is upper triangular Orthogonal Matrix: this is a square matrix whose columns are orthonormal vectors Positive Definite Matrix: one that satisfies x T Ax >, 8 x 6= omment: it is always assumed that A is symmetric Square Matrix: n n matrix (so, the number of rows equals the number of columns) Strictly Diagonal Dominant Matrix: one that satisfies nx a ii > a ij 8i j= j6=i Symmetric Matrix: one that satisfies A T = A. Worth mentioning: a symmetric matrix must be square. Tridiagonal Matrix: a square matrix with the only nonzero entries appearing on the diagonal, sub-diagonal and the super-diagonal Upper Triangular Matrix: one in which all entires below the diagonal are zero Worth mentioning: if A is upper triangular then A T is lower triangular E..4 Vector Products Inner (or, Dot) product: assuming v, w 2 R n,thenv w P n i= v iw i. This can be written in vector form as v w = v T w or as v w = w T v. Outer product: assuming v 2 R n and w 2 R m,then vw T " " " Bw v w 2 v w m # # # A. Worth mentioning: a) vw T is an n m matrix; b) if v and w are nonzero, then vw T has rank one; c) (vw T ) T = wv T ; and d) vv T is a symmetric matrix 3 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
4 E..5 Useful Matrix Theorems Orthogonal Matrix Theorem. Suppose Q is an n n matrix.. Q is an orthogonal matrix if and only if Q = Q T. 2. Q is an orthogonal matrix if and only if its rows are orthonormal vectors. 3. If Q is an orthogonal matrix then Qx 2 = x If Q and Q 2 are orthogonal matrices then Q Q 2 is an orthogonal matrix. 5. If Q is an orthogonal matrix, then Q T is an orthogonal matrix. Note that (Q T ) is the same matrix as (Q ) T, and both can be designated as Q T. 6. If Q is an orthogonal matrix, then det(q) =±. Positive Definite Matrix Theorem. Assume A is a symmetric n n matrix.. A is not positive definite if any diagonal entry is negative or zero, or if the largest number, in absolute value, is o the diagonal. 2. A is positive definite if the diagonals are all positive and it is strictly diagonal dominant. 3. A is positive definite if and only if all of its eigenvalues are positive. 4. If A is positive definite then it is invertible and A is positive definite. A T A and AA T Theorem. Assume A is an m n matrix.. A T A is an n n symmetric matrix with non-negative eigenvalues, and AA T is an m m symmetric matrix with non-negative eigenvalues. 2. If the columns of A are linearly independent, then A T A is positive definite. Moreover, apple 2 (A T A)=[apple 2 (A)] If the rows of A are linearly independent, then AA T is positive definite. Moreover, apple 2 (AA T )= apple 2 (A T ) If is a nonzero eigenvalue of A T A with eigenvector x, then is an eigenvalue of AA T with eigenvector Ax. If =is an eigenvalue of A T A with eigenvector x, and if Ax 6=, then =is an eigenvalue of AA T with eigenvector Ax. 5. If is a nonzero eigenvalue of AA T with eigenvector y, then is an eigenvalue of A T A with eigenvector A T y.if =is an eigenvalue of AA T with eigenvector y, and if A T y 6=, then =is an eigenvalue of A T A with eigenvector A T y. 4 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
5 E.2 Eigenvalue Problem Given a square matrix A, find the number(s) and nonzero vectors x that satisfy Ax = x. Analytical procedure to solve this problem:. Solve (for ) det(a I) = This is known as the characteristic equation. 2. For each eigenvalue, solve (for x) (A I)x = E.2. Properties and Formulas for Eigenvalues and Eigenvectors Defective matrix: An n n matrix is said to be defective if it does not have n linearly independent eigenvectors. Geometric multiplicity: The number of linearly independent eigenvectors for an eigenvalue is the eigenvalue s geometric multiplicity. Inverse Matrix: a matrix with a zero eigenvalue is not invertible. Nonuniqueness: If x is an eigenvector, then x is an eigenvector for the same eigenvalue for any nonzero scalar. Rayleigh quotient: given an eigenvector x, then the associated eigenvalue can be computing using the formula = x Ax x x Trace formula: Let A is an n n matrix, with eigenvalues, 2,, n (each eigenvalue listed as many times as its respective geometric multiplicity), then tr(a) = nx i= i, 5 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
6 E.2.2 Useful Eigenvalue Theorems Symmetric Eigenvalue Theorem. If A is a symmetric n n matrix, then the following hold:. Its eigenvalues are real numbers. 2. If x i and x j are eigenvectors for di erent eigenvalues, then x i x j =. 3. It is possible to find a set of orthonormal basis vectors u, u 2,, u n, where each u i is an eigenvector for A. Modified Eigenvalue Theorem. Suppose the eigenvalues of an n n matrix A are, 2,, n.. Setting B = A!I, where! is a constant, then the eigenvalues of B are!, 2!,, n!. Also, if x i is an eigenvector for A that corresponds to i, then x i is an eigenvector for B that corresponds to i!. 2. If A is invertible, then the eigenvalues of A are /, / 2,, / n. Also, if x i is an eigenvector for A that corresponds to i, then x i is an eigenvector for A that corresponds to / i. Spectral Decomposition Theorem. If A is a symmetric matrix, then it is possible to factor A as A = QDQ T, where D is a diagonal matrix and Q is an orthogonal matrix. Letting u, u 2,, u n be orthonormal eigenvectors for A, with corresponding eigenvalues, 2,, n, then D = and the ith column of Q is u i. Diagrammatically, the factorization can be written as " " " A = Bu u 2 u n # # # A@ Worth mentioning: The factorization can be written as n A, n B A@ A = W + 2 W n W n, where W i = u i u T i. This is an example of an outer product expansion. u! u 2!. u n! A. 6 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
7 Outer Product Theorem.. If A is n p and B is p m, then their product has the outer product expansion AB = a b T + a 2 b T a p b T p, where a j is the jth column from A and b j is the jth column from B T. 2. If v, w 2 R n are nonzero, then A = vw T has exactly two eigenvalues: = v w, and 2 =. For, v is an eigenvector (and there are no other linearly independent eigenvectors for ). The eigenvalue 2 has a geometric multiplicity of n and each of its eigenvectors is orthogonal to v. 3. Assume that A is n n, and it can be written as A = u u T + 2 u 2 u T k u k u T k, where the i s are nonzero and apple k apple n. In the u i s are orthonormal, then each u i is an eigenvector for A with corresponding eigenvalue i. Ifk<n, then the only other eigenvalue for A is =, and it has a geometric multiplicity of n k. Alsonotethat A is symmetric. Worth mentioning: Result 3 is the converse of the Spectral Decomposition Theorem. 7 Introduction to Scientific omputing and Data Analysis by M. Holmes (Springer, 26)
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