Review of Chapter 2

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.

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Subsubsection 2.11.1.2 Solving Process:

Learn the exact Gaussian Elimination algorithm:

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

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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: 🔗: \(\displaystyle \mathbf{d}= P \bb\)
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Forwardsolve: 🔗: \(\displaystyle L \yb = \mathbf{d}\)
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Backsolve: 🔗: \(\displaystyle U \xb = \yb\)
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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: 🔗: \(\displaystyle \det(A - \lambda I) = 0\)
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Equation of the eigenvector: 🔗: \(\displaystyle (A - \lambda I) \vb = \mathbf{0}\)
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Residual for an approximate eigenvector-eigenvalue pair: 🔗: \(\displaystyle r = \| A \vb - \lambda \vb \|\)
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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: \(\wb = \ub \pm i \vb\text{.}\)

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Subsubsection 2.11.1.7 Vibrational modes:

Eigenvalues are frequencies squared. Eigenvectors represent modes.

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Subsubsection 2.11.1.8 Power Method:

Repeatedly multiply \(\xb\) by \(A\) and divide by the element with the largest absolute value.
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The element of largest absolute value converges to largest absolute eigenvalue.
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The vector converges to the corresponding eigenvector.
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Convergence assured for a real symmetric matrix, but not for an arbitrary matrix, which may not have real eigenvalues at all.
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Subsubsection 2.11.1.9 Inverse Power Method:

Apply power method to \(A^{-1}\text{.}\)
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Use solving rather than the inverse.
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If \(\lambda\) is an eigenvalue of \(A\) then \(1/\lambda\) is an eigenvalue for \(A^{-1}\text{.}\)
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The eigenvectors for \(A\) and \(A^{-1}\) are the same.
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Subsubsection 2.11.1.10 Symmetric and Positive definite:

Symmetric: \(A = A'\text{.}\)
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If \(A\) is symmetric its eigenvalues are real.
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Positive definite: \(A\xb \cdot \xb >0\text{.}\)
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If \(A\) is positive definite, then its eigenvalues are positive.
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Subsubsection 2.11.1.11 QR method (Not covered in MATH 3600 at Ohio University):

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

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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)`

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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.

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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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