Matrices Notes

Notes for Matrices

Suffix notation:

Suffix notation is a useful form of notation for dealing with tensors. It expresses these tensors in terms of their elements:

• a_i for a vector

• A_ij for a matrix

• α_ijk for a three-dimensional tensor

Operations involving tensors can therefore be expressed as summations over their elements, such that:

The summation convention is then applied, by which if indices are multiplied and repeated, they are summed over implicitly, such that the sigma notation is removed:

It is important to note that a_ib_i represents a scalar quantity. Thus, in order to specify a vector quantity such as a cross product in suffix notation, it is necessary to specify that the suffixes represent only the ith element of the resulting vector; (ab)_i.

Matrix multiplication in suffix notation:

In suffix notation, the product of two matrices can be understood and written as follows:

$$\mathbf{AB} = \sum_{i}^{}{\sum_{j}^{}\left( \mathbf{AB} \right)_{ij}} = \left( \mathbf{AB} \right)_{ij}$$

Introducing some arbitrary vector x, and noting that matrix multiplication is associative:

(AB)x=A(Bx)

[(AB)x]_i=[A(Bx)]_i=A_ik(Bx)_k=A_ikB_kjx_j

(AB)_ijx_j=A_ijB_kjx_j

$$\boxed{\mathbf{\therefore}\left( \mathbf{AB} \right)_{\mathbf{ij}}\mathbf{=}\mathbf{A}_{\mathbf{ik}}\mathbf{B}_{\mathbf{kj}}}$$

Where k is repeated and thus summed over implicitly.

This rule can be extended, via the same repeated reasoning, to four matrices:

$$\boxed{\left( \mathbf{ABCD} \right)_{\mathbf{ij}}\mathbf{=}\mathbf{A}_{\mathbf{ik}}\mathbf{B}_{\mathbf{kl}}\mathbf{C}_{\mathbf{lm}}\mathbf{D}_{\mathbf{mj}}}$$

The Kronecker delta:

The Kronecker delta is defined as:

$$\boxed{\delta_{ij} = \{\begin{matrix} 0,\ \ i \neq j \\ 1,\ \ i = j \\ \end{matrix}}$$

It can be expressed in terms of the vector differential operator:

$${\frac{\partial x_{j}}{\partial x_{i}}|}_{x_{k}} = \partial_{i}x_{j},\ \ k \neq j$$

Which is equal to 1 if i=j, but if ij then i equals k, and thus _ix_j=0.

$$\boxed{\therefore\delta_{ij} \equiv \partial_{i}x_{j}}$$

Where x is a vector.

Applying the Kronecker delta:

Applying the Kronecker delta to some vector v leads to the following result:

$$\boxed{\delta_{ij}\mathbf{v}_{\mathbf{j}}\mathbf{=}\mathbf{v}_{\mathbf{i}}}$$

Since if i=j,δ_ij=1 but if ij,δ_ij=0. Thus the Kronecker delta is often said to ‘change the suffix’ of a vector.

The Levi-Civita symbol:

The Levi-Civita symbol is defined as:

$$\boxed{\epsilon_{ijk} = \{\begin{matrix} 1,\ \ ijk = cyclic\ permutation \\ - 1,\ \ ijk = antcyclic\ permutation \\ 0,\ \ ijk = any\ equal\ indices \\ \end{matrix}}$$

It can thus be said that:

$$\boxed{\left( \mathbf{a} \times \mathbf{b} \right)_{i} = \epsilon_{ijk}a_{j}b_{k}}$$

Which can be verified in three dimensions as follows:

Let i=1:

(ab)₁=ϵ₁₂₃a₂b₃+ϵ₁₃₂a₃b₂+zeroes(equalindices)

(ab)₁=a₂b₃a₃b₂

Let i=2:

(ab)₂=ϵ₂₁₃a₁b₃+ϵ₂₃₁a₃b₁+zeroes(equalindices)

(ab)₁=a₁b₃+a₃b₁

Let i=3:

(ab)₃=ϵ₃₁₂a₁b₂+ϵ₃₂₁a₂b₁+zeroes(equalindices)

(ab)₃=a₁b₂+a₂b₁

Relation between δ_ij and ϵ_ijk:

The following relation exists between the two operators:

$$\boxed{\mathbf{\epsilon}_{\mathbf{ijk}}\mathbf{\epsilon}_{\mathbf{ilm}}\mathbf{=}\mathbf{\delta}_{\mathbf{jl}}\mathbf{\delta}_{\mathbf{km}}\mathbf{-}\mathbf{\delta}_{\mathbf{jm}}\mathbf{\delta}_{\mathbf{kl}}}$$

Which can be remembered as ‘same minus different’. That is, δ_jl has both the second indices j and l, δ_km has both the third indices k and m, and δ_jm and δ_kl have mixed ‘different’ indices.

The inner/scalar product:

The inner/scalar product is defined for any kind of vector space, but is not part of the axioms which define vector spaces. In three dimensions, the inner/scalar product for real and complex vectors and matrices is defined in the following ways (in suffix notation):

Properties of the inner/scalar product:

The scalar product has the following properties:

1. x•x=x_ix_i=|x_i|², and is real and positive. It defines the magnitude of x as:

$$\left| \mathbf{x} \right| = \sqrt{\mathbf{x \bullet x}}$$

2. If x•y=0, x and y are said to be orthogonal. For column vectors in three dimensions, this means that the vectors are perpendicular, but this inner product definition is much broader, encompassing not only higher-order tensors, but also functions.

3. A basis {e₁,e₂,…,e_n} which satisfies e_i•e_i=1 and e_i•e_j=0 for ij is said to be orthonormal, and can be written in terms of the Kronecker delta as:

e_i•e_j=δ_ij

Basic nomenclature:

The following are basic properties of matrices:

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