• #Chapter 2: Basic Linear Algebra for Tensors
  • #(2.1) Introduction :-
  • #(2.2) Tensor notation for matrixes, vectors & determinants
  • #Vectors :-
  • #Basic Formulas
  • #(1) matrix multiplication
  • #(2) Identity matrix
  • #(3) Inverse of a square matrix :-
  • #(4) Transpose of a matrix :-
  • #(5) orthogonal matrix :-
  • #Permutation symbol :
  • #determinent of square matrix :-
  • #laplace expansion of a determinent :-
  • #Scalar product of vectors :-
  • #norm/length of a vector :-
  • #angle b/w two vectors :-
  • #vector product/cross product in R^3 :-
  • #(2.3) Inverting a matrix :-

tensors | Tensors chapter 1

[Page heading as written : (Chapter -2) (Basic Linear Algebra for Tensors)]

[A struck-out label "(1.15)" precedes "(2.1)" in the margin.]

The main purpose is to reformulate the expressions of linear algebra and matrix theory using summation convention.

\((a_{ij})\) represents an element of matrix

\(i \rightarrow\) row number \(\quad\) (1 to \(m\))

\(j \rightarrow\) column number \(\quad\) (1 to \(n\))

A fuller notation is \([a_{ij}]_{mn}\)

upper index matrix notation

\[ [a^i_j]_{mn} \equiv \begin{bmatrix} a^1_1 & a^1_2 & a^1_3 & -- & a^1_n\\ a^2_1 & a^2_2 & & & a^2_n\\ a^3_1 & a^3_2 & ---- & & a^3_n\\ \vdots & & & & \\ a^m_1 & a^m_2 & -- & & a^m_n \end{bmatrix} \] \[ [a^{ij}]_{mn} \equiv \begin{bmatrix} a^{11} & a^{12} & a^{13} & .. & a^{1n}\\ a^{21} & a^{22} & a^{23} & .. & a^{2n}\\ \vdots & & & & \\ a^{m1} & a^{m2} & a^{m3} & -- & a^{mn} \end{bmatrix} \]

(ex 2.1)

\[ [c^i_j]_{23} = \begin{bmatrix} c^1_1 & c^1_2 & c^1_3\\ c^2_1 & c^2_2 & c^2_3 \end{bmatrix} \] \[ [d^j_i]_{23} = \begin{bmatrix} d^1_1 & d^1_2 & d^1_3\\ d^2_1 & d^2_2 & d^2_3 \end{bmatrix} \equiv [d^i_j]_{23} \] \[ [x^r_s]_{14} = \begin{bmatrix} x^1_1 & x^1_2 & x^1_3 & x^1_4 \end{bmatrix} \] \[ [y^{pq}]_{42} = \begin{bmatrix} y^{11} & y^{12}\\ y^{21} & y^{22}\\ y^{31} & y^{32}\\ y^{41} & y^{42} \end{bmatrix} \]

A real n-diamensional vector is any column matrix \(V = [x_{ij}]_{n1}\) with real components \((x_i = x_{i1})\)

(or) we can simply write \((V = (x_i))\). The collection of all real n-diamentional vectors is the n-diamensional real vector space denoted \(R^n\).

(i.e. \(R^n\) is made up of all real n-d vectors.)

Vector sums are determined by coordinate wise add\(^n\) as are matrix sums.

if \(A \equiv [a_{ij}]_{mn}\) & \(B \equiv [b_{ij}]_{mn}\)

Then \(A+B \equiv [a_{ij} + b_{ij}]_{mn}\)

Scalar multiplication of a vector or matrix is defined by \(\lambda [a_{ij}]_{mn} \equiv [\lambda a_{ij}]_{mn}\)

\[ A \equiv [a_{ij}]_{mn} \quad \& \quad B \equiv [b_{ij}]_{nk} \quad \text{then} \] \[ AB = [a_{ir} b_{rj}]_{mk} \]

\(\hookrightarrow\) represents summation over '\(r\)'.

For mixed / upper indices :-

\[ AB \equiv [a^i_j]_{mn}\,[b^i_j]_{nk} \qquad \Big| \qquad AB \equiv [a^{ij}]_{mn}\,[b^{ij}]_{nk} \] \[ AB = [a^i_r\, b^r_j]_{mk} \qquad \Big| \qquad AB = [a^{ir}\, b^{sj}]_{mk} \]

In terms of kronecker delta, identity matrix of order '\(n\)' is

\[ I = [\delta_{ij}]_{nn} \equiv [\delta^i_j]_{nn} = [\delta^{ij}]_{nn} \]

which has property \((IA = AI = A)\) for \(A = [a^{ij}]_{nn}\)

or \([a_{ij}]_{nn}\)

or \([a^i_j]_{nn}\)

A square matrix \(A = [a_{ij}]_{nn}\) is invertible if \(\exists\) a unique matrix \(B = [b_{ij}]_{nn}\) called inverse of \(A\) such that \((AB = BA = I)\)

In terms of components criterion reads

\[ AB = BA = I \] \[ (a_{ir} b_{rj} = b_{ir} a_{rj} = \delta_{ij}) \]

(or) For mixed or upper indices

\[ a^i_r b^r_j = b^i_r a^r_j = \delta^i_j \] \[ a^{ir} b^{rj} = b^{ir} a^{rj} = \delta^{ij} \]

Transpose of arbitrary matrix is

\[ A^T \equiv [a_{ij}]^T_{mn} = [a_{ji}]_{nm} \]

if \(a_{ij} = a_{ji}\) \(\quad\) (\(A\) is symmetric matrix

\[ a_{ij} = -a_{ji} \qquad A \ \text{is skew-symmetric} \]

or Anti\(^o\)-symmetric

A matrix '\(A\)' is orthogonal if \(A^T = A^{-1}\)

or if \(A^T A = A A^T = I\)

\[ (A^T = A^{-1}) \qquad\qquad A = [a_{ij}]_{nn} \] \[ a_{jp} = b_{ij} \qquad\qquad (A^{-1}) = b_{ij} \] \[ e_{ijk\cdots\omega} = 0 \qquad \left(\text{if any two subscripts are same}\right. \] \[ = (-1)^P \]

\(P = \) no. of subscript transpositions (interchg) required to bring \((i,j,k\cdots\omega)\) to natural order \((1,2,3,\cdots n)\)

If \(A \equiv [a_{ij}]_{nn}\) is any square matrix

then

\[ \det A = e_{i_1 i_2 i_3 \cdots i_n}\ a_{1i_1}\, a_{2i_2}\, a_{3i_3} -- a_{ni_n} \]

(ex) for \(2\times2\) matrix

\[ A = \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix} \] \[ |A| = e_{i_1 i_2}\, a_{1i_1}\, a_{2i_2} \]

repeated indices get summed over

So,

\[ |A| = e_{1i_2}\, a_{11}\, a_{2i_2} + e_{2i_2}\, a_{12}\, a_{2i_2} \] \[ = a_{11}\left(e_{1i_2}\, a_{2i_2}\right) + a_{12}\left(e_{2i_2}\, a_{2i_2}\right) \] \[ = a_{11}\left(\underbrace{e_{11}}_{0}\, a_{21} + e_{12}\, a_{22}\right) + a_{12}\left(e_{21}\, a_{21} + \underbrace{e_{22}\, a_{22}}_{0}\right) \] \[ = e_{12}\, a_{11} a_{22} + e_{21}\, a_{12} a_{21} \] \[ |A| = a_{11} a_{22} - a_{12} a_{21} \] \[ |AB| = |A||B| \quad \text{and} \quad |A^T| = |A| \]

let \(M_{ij}\) be minor of \(a_{ij}\) in \(A\) and \(C_{ij}\) be cofactor of \(a_{ij}\) in \(A\).

\[ C_{ij} = (-1)^{i+j}\, M_{ij} \] \[ |A| = a_{1j} C_{1j} = a_{2j} C_{2j} = a_{3j} C_{3j} = \cdots a_{nj} C_{nj} \] \[ |A| = a_{i1} C_{i1} = a_{i2} C_{i2} = a_{i3} C_{i3} \cdots = a_{in} C_{in} \]

\(\rightarrow\) repeated indices gets summed over

\[ \boxed{|A| = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}} \]

If \(u = (x_i)\)

\(\qquad v = (y_i)\)

\[ u\cdot v = u^T v = x_i y_i = x_1 y_1 + x_2 y_2 + x_3 y_3 + \cdots + x_n y_n \] \[ u^2 = u\cdot u = x_i x_i \] \[ u\cdot v = 0 \ \Rightarrow \ \bar{u}\ \&\ \bar{v}\ \text{are orthogonal.} \quad (x_i y_i = 0) \]

for \(\bar{u} = (x_i)\)

\[ \|u\| = \sqrt{x_i x_i} \]

angle b/w two vectors \(u = (x_i)\) & \(V = (y_i)\) is defined by

\[ \cos\theta = \frac{u v}{|u||v|} = \frac{x_i y_i}{\sqrt{x_j x_j}\ \sqrt{y_k y_k}} \]

for \(\theta = \pi/2\) we need \((x_i y_i = u\cdot v = 0)\).

\(u = (x_i)\)

\(v = (y_i)\)

\[ u \times v = \hat{i}\left(x_2 y_3 - x_3 y_2\right) - \hat{j}\left(x_1 y_3 - x_3 y_1\right) + \left(x_1 y_2 - y_1 x_2\right)\hat{k} \] \[ \boxed{(\bar{u}\times\bar{v})_i = e_{ijk}\, x_j\, y_k} \]

or

\[ \left((\bar{u}\times\bar{v})_i = \epsilon_{ijk}\, x_j\, y_k\right) \]

A square matrix is invertible if there exist \(A^{-1}\) such that

\[ (A A^{-1} = A^{-1} A = I) \]

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