Methods III Tripos Questions

Linear Algebra

Eigenvalues and Eigenvectors

Find eigenvalues and eigenvectors of $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$.

Characteristic polynomial:

$\det(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10$

$= (\lambda - 2)(\lambda - 5) = 0$

Eigenvalues: $\lambda_1 = 2$, $\lambda_2 = 5$.

For $\lambda_1 = 2$:

$(A - 2I)\mathbf{v} = \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}\mathbf{v} = \mathbf{0}$

$v_1 + v_2 = 0$, so $\mathbf{v}_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$

For $\lambda_2 = 5$:

$(A - 5I)\mathbf{v} = \begin{pmatrix} -1 & 2 \\ 1 & -2 \end{pmatrix}\mathbf{v} = \mathbf{0}$

$-v_1 + 2v_2 = 0$, so $\mathbf{v}_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$

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Diagonalisation

Diagonalise $A$ from the previous example.

$S = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}$

$S^{-1} = \frac{1}{3}\begin{pmatrix} 1 & -2 \\ 1 & 1 \end{pmatrix}$

$\Lambda = \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$

Verify: $S\Lambda S^{-1} = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} = A$

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Determinant Properties

Show that if $A$ and $B$ are $n \times n$, then $\det(AB) = \det(A)\det(B)$.

For $2 \times 2$ matrices:

$\det(AB) = (a_{11}b_{11} + a_{12}b_{21})(a_{21}b_{12} + a_{22}b_{22}) - (a_{11}b_{12} + a_{12}b_{22})(a_{21}b_{11} + a_{22}b_{21})$

Expanding and collecting terms gives $(a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21}) = \det(A)\det(B)$.

(General proof uses properties of multilinear forms.)

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Partial Differential Equations

Laplace Equation

Solve $\nabla^2 \phi = 0$ in $0 < x < \pi$, $0 < y < \pi$ with:

• $\phi(0, y) = 0$
• $\phi(\pi, y) = 0$
• $\phi(x, 0) = 0$
• $\phi(x, \pi) = \sin x$

By separation: $\phi = X(x)Y(y)$

From $x$ boundaries: $X = \sin(nx)$, $n = 1, 2, 3, ...$

From $y = 0$ boundary: $Y(0) = 0$, so $Y = \sinh(ny)$.

Solution:

$\phi = \sum_{n=1}^{\infty} C_n \sin(nx)\sinh(ny)$

At $y = \pi$: $\phi(x, \pi) = \sin x$

This matches only $n = 1$: $C_1\sinh(\pi) = 1$, so $C_1 = 1/\sinh(\pi)$.

$\phi = \frac{\sin(x)\sinh(y)}{\sinh(\pi)}$

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Heat Equation

Solve $u_t = u_{xx}$ on $0 < x < 1$, $t > 0$ with $u(0, t) = u(1, t) = 0$ and $u(x, 0) = \sin(2\pi x)$.

By separation: $u = X(x)T(t)$

From boundaries: $X_n = \sin(n\pi x)$, $n = 1, 2, 3, ...$

$T_n(t) = e^{-n^2\pi^2 t}$

All $C_n = 0$ except $C_2$ (matching initial condition).

$u(x, t) = \sin(2\pi x)e^{-4\pi^2 t}$

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Wave Equation

Solve $y_{tt} = y_{xx}$ on $0 < x < 1$ with:

• $y(0, t) = y(1, t) = 0$ (fixed ends)
• $y(x, 0) = \sin(\pi x)$
• $\dot{y}(x, 0) = 0$

Solution:

$y(x, t) = \sin(\pi x)\cos(\pi t)$

(Standing wave with $n = 1$.)