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Section 2.11 Review of Chapter 2

Subsection 2.11.1 Methods and Formulas

Subsubsection 2.11.1.1 Basic Matrix Theory:

Identity matrix: \(A I = A\text{,}\) \(I A = A\text{,}\) and \(I \vb = \vb\)
Inverse matrix: \(A A^{-1}= I\) and \(A^{-1}A = I\)
Norm of a matrix: \(\|A\| \equiv \max_{\|\vb\|=1}\|A\vb\|\)
A matrix may be singular or nonsingular. See Section 2.3.

Subsubsection 2.11.1.2 Solving Process:

Learn the exact Gaussian Elimination algorithm:
  • Row \(j\) \(\mapsto\) Row \(j\) - (ratio) Row \(i\)
  • Gaussian Elimination this way produces LU decomposition
  • Row Pivoting (bigger absolute number on top).
  • Back Substitution.

Subsubsection 2.11.1.3 Condition number:

\begin{equation*} \textrm{cond}(A) \equiv \max \left( \frac{ \|\delta \xb\|/\|\xb\|}{ \frac{\|\delta A\|}{\|A\|} + \frac{\|\delta \bb\|}{\|\bb\|} }\right) = \max \left( \frac{\text{Relative error of output}}{\text{Relative error of inputs}}\right). \end{equation*}
A big condition number is bad; in engineering it usually results from poor design.

Subsubsection 2.11.1.4 LU factorization:

The LU factorization is a by-product of Gaussian Elimination (if done with the correct algorithm).
\begin{equation*} PA = LU\text{.} \end{equation*}
Solving steps:
Multiply by P:
\(\displaystyle \mathbf{d}= P \bb\)
Forwardsolve:
\(\displaystyle L \yb = \mathbf{d}\)
Backsolve:
\(\displaystyle U \xb = \yb\)

Subsubsection 2.11.1.5 Eigenvalues and eigenvectors:

A nonzero vector \(\vb\) is an eigenvector and a number \(\lambda\) is its eigenvalue if
\begin{equation*} A \vb = \lambda \vb\text{.} \end{equation*}
Characteristic equation:
\(\displaystyle \det(A - \lambda I) = 0\)
Equation of the eigenvector:
\(\displaystyle (A - \lambda I) \vb = \mathbf{0}\)
Residual for an approximate eigenvector-eigenvalue pair:
\(\displaystyle r = \| A \vb - \lambda \vb \|\)

Subsubsection 2.11.1.6 Complex eigenvalues:

Occur in conjugate pairs: \(\lambda_{1,2}= \alpha \pm i \beta\) and eigenvectors must also come in conjugate pairs: \(\wb = \ub \pm i \vb\text{.}\)

Subsubsection 2.11.1.7 Vibrational modes:

Eigenvalues are frequencies squared. Eigenvectors represent modes.

Subsubsection 2.11.1.8 Power Method:

  • Repeatedly multiply \(\xb\) by \(A\) and divide by the element with the largest absolute value.
  • The element of largest absolute value converges to largest absolute eigenvalue.
  • The vector converges to the corresponding eigenvector.
  • Convergence assured for a real symmetric matrix, but not for an arbitrary matrix, which may not have real eigenvalues at all.

Subsubsection 2.11.1.9 Inverse Power Method:

  • Apply power method to \(A^{-1}\text{.}\)
  • Use solving rather than the inverse.
  • If \(\lambda\) is an eigenvalue of \(A\) then \(1/\lambda\) is an eigenvalue for \(A^{-1}\text{.}\)
  • The eigenvectors for \(A\) and \(A^{-1}\) are the same.

Subsubsection 2.11.1.10 Symmetric and Positive definite:

  • Symmetric: \(A = A'\text{.}\)
  • If \(A\) is symmetric its eigenvalues are real.
  • Positive definite: \(A\xb \cdot \xb >0\text{.}\)
  • If \(A\) is positive definite, then its eigenvalues are positive.

Subsubsection 2.11.1.11 QR method (Not covered in MATH 3600 at Ohio University):

  • Transform \(A\) into \(H\) the Hessian form of \(A\text{.}\)
  • Decompose \(H\) in \(QR\text{.}\)
  • Multiply \(Q\) and \(R\) together in reverse order to form a new \(H\text{.}\)
  • Repeat
  • The diagonal of \(H\) will converge to the eigenvalues of \(A\text{.}\)

Subsection 2.11.2 MATLAB

Subsubsection 2.11.2.1 Matrix arithmetic:

A = [ 1 3 -2 5 ; -1 -1 5 4 ; 0 1 -9 0] Manually enter a matrix.
u = [ 1 2 3 4]'
A*u
B = [3 2 1; 7 6 5; 4 3 2]
B*A multiply \(B\) times \(A\text{.}\)
2*A multiply a matrix by a scalar.
A + A add matrices.
A + 3 add 3 to every entry of a matrix.
B.*B component-wise multiplication.
B.^3 component-wise exponentiation.

Subsubsection 2.11.2.2 Special matrices:

I = eye(3) identity matrix
D = ones(5,5)
O = zeros(10,10)
C = rand(5,5) random matrix with uniform distribution in \([0,1]\text{.}\)
C = randn(5,5) random matrix with normal distribution.
hilb(6)
pascal(5)

Subsubsection 2.11.2.3 General matrix commands:

size(C) gives the dimensions (\(m \times n\)) of \(A\text{.}\)
norm(C) gives the norm of the matrix.
det(C) the determinant of the matrix.
max(C) the maximum of each row.
min(C) the minimum in each row.
sum(C) sums each row.
mean(C) the average of each row.
diag(C) just the diagonal elements.
inv(C) inverse of the matrix.
C' transpose of the matrix.

Subsubsection 2.11.2.4 Matrix decompositions:

[L U P] = lu(C)
[Q R] = qr(C)
H = hess(C) transform into a Hessian tri-diagonal matrix, which has the same eigenvalues as \(A\text{.}\)
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