Matrices
Vector space
Definition
- A set of elements on which vector addition and scalar multiplication are defined
- Closed under these operations
- Includes identity \(\mathbf{0}\) for vector addition
Note:
- 2D spaces \(\mathbb{R}^2 \neq \mathbb{C}\), since \(\mathbb{C}\) has a single point of \(\infty\).
Span
The set of all vectors that are linear combinations of any vectors of a set \(\mathbb{S}\).
Linearly independent
None of the vectors is a linear combination of others, or \[ x_1\mathbf{e}_1+x_2\mathbf{e}_2+x_3\mathbf{e}_3+...=0 \iff x_1=x_2=x_3=...=0 \]
Basis
A set that spans \(\mathbb{V}\) and are linearly independent.
- The number of elements is the dimension of \(\mathbb{V}\).
- Any vector can be expressed as a linear combination of the basis set in a unique way.
- Can have infinite dimensions - e.g. Fourier series
Change of bases
Primed / unprimed opposite to lecture notes convention.
- Idea: the same vector has different components in different bases.
\[ \mathbf{e}_j'=\mathbf{e}_i\mathbf{R}_{ij} \]
where \(\mathbf{R}_{ij}\) is the \(i\)th component of \(\mathbf{e}_j'\) in the unprimed basis, i.e. \[ \mathbf{R}_{ij}= \begin{pmatrix} \uparrow & \uparrow & \uparrow\\ \mathbf{e_1'} & \mathbf{e_2'} &\mathbf{e_3'}\\ \downarrow &\downarrow &\downarrow \end{pmatrix} \] Check by reversing the argument \[ \begin{pmatrix} \uparrow & \uparrow & \uparrow\\ \mathbf{e_1'} & \mathbf{e_2'} &\mathbf{e_3'}\\ \downarrow &\downarrow &\downarrow \end{pmatrix} \begin{pmatrix} \uparrow \\ \mathbf{e'}\\ \downarrow \end{pmatrix} =\begin{pmatrix} \uparrow \\ \mathbf{e'}\,\mathrm{in}\,\mathbf{e}\\ \downarrow \end{pmatrix} \] Now, \[ \mathbf{x}=x_i\mathbf{e}_i=x_j'\mathbf{e}_j'=x_j'\mathbf{e}_i\mathbf{R}_{ij}\\ \implies x_i=\mathbf{R}_{ij}x_j' \]
Inverse
Define the transformations \[ \mathbf{e}_j'=\mathbf{e}_i\mathbf{R}_{ij} \\ \mathbf{e}_j=\mathbf{e}_i'\mathbf{S}_{ij} \] Sub in, \[ \mathbf{e}_j=\mathbf{e}_k\mathbf{R}_{ki}\mathbf{S}_{ij} \] True iff \(\mathbf{R}_{ki}\mathbf{S}_{ij}=\delta_{kj}\), similarly, \(\mathbf{S}_{ki}\mathbf{R}_{ij}=\delta_{kj}\). Hence, inverse transform is represented by inverse matrix.
Transforming matrices
\[\begin{align} \mathbf{A}\mathbf{x}=\mathbf{e}_i\mathbf{A}_{ij}x_j&=\mathbf{e}_i'\mathbf{A'}_{ij}x_j'\\ \mathbf{e}_i\mathbf{A}_{ij}x_j&=\mathbf{e}_k\mathbf{R}_{ki}\mathbf{A'}_{ij}x_j'\\ \mathbf{A}\mathbf{x}&=\mathbf{R}\mathbf{A'}\mathbf{R}^{-1}\mathbf{x'}\\ \mathbf{A}&=(\mathbf{R}\mathbf{A'})\mathbf{R}^{-1}\\ \end{align}\]
Linear Operation
\[ \mathbf{A}(\alpha\mathbf{x}+\mathbf{y})=\alpha\mathbf{A}\mathbf{x}+\mathbf{A}\mathbf{y} \]
Without reference to any bases.
Inner Product
\[ <\!\!x|y\!\!>=x_i^*y_i \]
- Linear in 2nd argument
- Anti-linear in 1st argument
- Hermitian: \(<\!\!x|y\!\!>^*=<y|x>\)
- Positive-definite: \(<\!\!x|x\!\!>\geq0\)
Inner Product and bases
\[ <\!\!x|y\!\!>=<\!\!x_i\mathbf{e}_i|y_j\mathbf{e}_j\!\!>=x_i^*y_j<\!\!\mathbf{e}_i|\mathbf{e}_j\!\!>=x_i^*y_j\mathbf{G}_{ij} \]
- Metric coefficient: \(\mathbf{G}_{ij}\), hermitian
Cauchy-Schwartz
\[ |<\!\!\mathbf{x}|\mathbf{y}\!\!>|\leq||\mathbf{x}||\,||\mathbf{y}|| \]
Proof by considering \(<\!\!\mathbf{x}-\alpha\mathbf{y}|\mathbf{x}-\alpha\mathbf{y}\!\!>\), then choose appropriate phase of \(\alpha\), so that \(\alpha<\!\!\mathbf{x}|\mathbf{y}\!\!>\) is real and positive; appropriate length of \(\alpha=\frac{|\mathbf{x}|}{|\mathbf{y}|}\).
Hermitian Conjugate
\[ \mathbf{A}^\dagger=(\mathbf{A}^T)^*\\ \mathbf{AB}^\dagger=\mathbf{B}^\dagger\mathbf{A}^\dagger \]
An operator is adjoint if \(<\!\!\mathbf{A}^\dagger \mathbf{x}|\mathbf{y}\!\!>=<\!\!\mathbf{x}|\mathbf{A} \mathbf{y}\!\!>\).
Special square matrices
| Type | Relation |
|---|---|
| Real Symmetric | \(A_{ij}=A_{ji}\) |
| Real Antisymmetric | \(A_{ij}=-A_{ji}\) |
| Orthogonal | \(\mathbf{A}^\mathrm{T}=\mathbf{A}^{-1}\) |
| Hermitian | \(\mathbf{A}^\dagger=\mathbf{A}\) |
| Anti-hermitian | \(\mathbf{A}^\dagger=-\mathbf{A}\) |
| Unitary | \(\mathbf{A}^\dagger=\mathbf{A}^{-1}\) |
| Normal | \(\mathbf{A}^\dagger\mathbf{A}=\mathbf{A}\mathbf{A}^\dagger\) |
Special results
- If A is Hermitian then iA is anti-Hermitian
- if A is Hermitian then exp(iA) is unitary
c.f. real number
Eigenvectors
Two things to remember: \[ \mathbf{Ax}=\lambda\mathbf{x}\\ \mathbf{AS}=\mathbf{S\Lambda} \]
- Always find Eigenvalues first, by solving characteristic equation \(\det{(\mathbf{A}-\lambda\mathbf{I})}=0\).
- Roots
- n distinct roots - n linearly independent vectors
- m repeated roots - may be 1 to m vectors corresponding to it, and any linear combinations of those are Eigenvectors.
Derivation of Eigenvalue, Eigenvector properties of special matrices.
Diagonalisation
Diagonalisation is also a similarity transformation, into the Eigenvector basis.
- Only diagonalisable if the matrix has n independent eigenvectors.
Normal matrices can be diagonalised by unitary transformation. - Transformation between orthonormal bases is described by a unitary vector.
- Tr and Det are most easily studied in the diagonal form, invariant under similarity transformations.
Forms
Quadratic form
Any homogeneous quadratic funtion is a quadratic form of a symmetric matrix. \[ Q(\mathbf{x})=\mathbf{x}^T\mathbf{A}\mathbf{x}=A_{ij}x_i x_j \] By diagonalisation, we can get a \(Q(\mathbf{x}')\) with no cross-terms.
- The eigenvectors of A defines the principal axes of the quadratic form.
- Positive (semi-)definite means all eigenvalues are larger than (or equal to) 0.
Quadratic surface
\[ Q(\mathbf{x})=k \]
Ellipsoids - \(\lambda\) same sign
Hyperboloids - \(\lambda\) different signs
\(\lambda_1=\lambda_2=\lambda_3\) - sphere
\(\lambda_1=\lambda_2\) - revolution about z’ (axis of symmetry)
\(\lambda_3=0\) - conic section translated along z’
Hermitian Form
Similar idea for complex vector space. \[ H(\mathbf{x})=\mathbf{x}^\dagger\mathbf{H}\mathbf{x}=\sum\lambda_i|x_i'|^2 \]
Stationary property
Define Rayleigh quotient \[ \lambda(\mathbf{x})=\frac{\mathbf{x}^\dagger\mathbf{A}\mathbf{x}}{\mathbf{x}^\dagger\mathbf{x}} \]
- \(\lambda\) is a scalar
- \(\lambda(\alpha\mathbf{x})=\lambda(\mathbf{x})\)
- When \(\mathbf{x}\) is eigenvector, \(\lambda\) is eigenvalue.
Rayleigh-Ritz
Eigenvalues of a matrix is the stationery values of the Rayleigh quotient.
Proof by considering \(\lambda(\mathbf{x}+\delta \mathbf{x})-\lambda(\mathbf{x})\).
Cartesian Tensor
Use the transformation matrix \(L\): \[ L_{ij}=\mathbf{e}_i'\cdot\mathbf{e}_j\\ \mathbf{L}= \begin{pmatrix} \leftarrow & \mathbf{e_1'} & \rightarrow\\ \leftarrow & \mathbf{e_2'} &\rightarrow\\ \leftarrow &\mathbf{e_3'} &\rightarrow \end{pmatrix} =\begin{pmatrix} \uparrow & \uparrow & \uparrow\\ ... & \mathbf{e_i}\,\mathrm{in}\,\mathbf{e_i}' &...\\ \downarrow &\downarrow &\downarrow \end{pmatrix} \\ \] So the transformation law for a vector is (only this form makes sense for me…) \[ v_i'=L_{ij}v_j= \begin{pmatrix} \uparrow & \uparrow & \uparrow\\ ... & \mathbf{e_i}\,\mathrm{in}\,\mathbf{e_i}' &...\\ \downarrow &\downarrow &\downarrow \end{pmatrix} \begin{pmatrix} \uparrow\\ \mathbf{v}\\\downarrow \end{pmatrix} \]
Definition in terms of transformation laws
Vector: a set of coefficents \(v_i\), defined wrt orthonormal basis \(\mathbf{e}_i\), such that \(v_i'\) wrt another basis \(\mathbf{e}_i'\) are given by \(v_i'=L_{ij}v_j\).
Axial vectors: \(v_i'=\det(\mathbf{L})L_{ij}v_j\)
Tensors: \(T_{ijk...}'=L_{ia}L_{jb}L_{kc}...T_{abc...}\)
Pseudo-tensor: \(T_{ijk...}'=\det(\mathbf{L})L_{ia}L_{jb}L_{kc}...T_{abc...}\)
Examples
Everything is checked by the transformation law above.
- Delta is a 2nd order tensor
- Epsilon is a 3rd order pseudo tensor
- Inertia tensor: \(I_{ij}=\int_V\rho(\mathbf{x})(x_kx_k\delta_{ij}-x_ix_i)\,dV\)
Derive from the angular momentum definition.
Operations
- Addition
- Outer product
- Inner product (contraction)
Note: Cross-product is defined using the epsilon symbol, so any cross product is a pseudo-tensor.
- Symmetric and antisymmetric properties are invariant.
- Contraction of symmetric & antisymmetric indices \(S_{ijk...}A_{ijr...} = 0\).
Proof by using symmetry properties once and relabel once.
Second-order tensors
Two new things
Antisymmetric (3D)
Only three degrees of freedom, hence can be represented by a vector. \[ A_{ij}=\epsilon_{ijk}\omega_k \] The vector \(\omega\) is a dual vector, defined by \[ \omega_k=\frac{1}{2}\epsilon_{klm}A_{lm} \]
Symmetric (3D)
Can be further decomposed, uniquely, into a traceless symmetric matrix and identity. \[ S=\tilde{S}+\frac{\mathrm{Tr}(S)}{3}I \]
Isotropic tensors
Tensors or pseudo-tensors with the same components in all frames.
- Scalars are isotropic
- No non-zero rank 1 isotropic tensor
- Rank 2 isotropic tensors are \(\lambda\delta_{ij}\)
Proof by (wlog) considering a rotation \(\pi/2\) by z-axis and by y-axis.
- Rank 3 isotropic tensors are \(\lambda\epsilon_{ijk}\)
- Rank 4 isotropic tensors are \(\lambda\delta_{ij}\delta_{kl}+\mu\delta_{ik}\delta_{jl}+\nu\delta_{il}\delta_{jk}\)
Applications
- Since invariant, we can pick the most convenient frame (e.g eigenvector frame).
- Integrals
Integral 1
\[ X_i=\int_{r\leq a}x_i\rho(r)\,dV \]
Relabel the integration variables as primed. Transform \(x_i'=R_{ij}x_j\) we find \(X_i=X_i'\). But only isotropic rank 1 tensor is \(\mathbf{0}\).
Integral 2
\[ K_{ij}=\int_{r\leq a}x_ix_j\rho(r)\,dV \]
This is also isotropic, so \(K_{ij}=\lambda\delta_{ij}\). \[ K_{ij}=\frac{1}{3}(\int_{r\leq a}r^2\rho(r)\,dV)\delta_{ij} \] Only need to do 1 integral.
Tensor fields
- \(\nabla\), with components \(\partial_i\), is a vector in Cartesian only.
One idea (maybe) new
- The derivative of a second-order tensor \(\sigma_{ij}\) is a rank 3 tensor field \(\partial_i\sigma_{jk}\).