Engineering Math

18년 1학기에 서울시립대학교에서 수강했던 Engineering Math를 제 필요에 의해 정리해봅니다. 영어 강의이고 제가 다시 참고하기 위한 글입니다. 귀찮은 부분 다 건너 뛰어요.

Matrix

  • \(A = (a_{ij})\) : collection of numbers. i: row, j: col

  • \(A = \pmatrix{ 1 & 2 & 3 \\ 4 & 5 & 6 }\)에서 1~4 방향이 col, 4~6 방향이 row. 그리고 각각이 element.

  • \(a_{11} = 1, a_{12} = 2, a_{13} = 3\) …
  • \(A\) : 2x3 matrix (#row x #col matrix)

연산

Transpose

\(A = (a_{ij}), A^T = (a_{ji})\).

associative: \((AB)^T = B^TA^T\), \((ABC)^T = C^TB^TA^T\).

Multiplication

\[(AB)_{ij} = \sum^n_{k=1} (A)_{ik}(B)_{kj}\]

In general, \(AB \neq BA\). -> matrix multiplication is non commutative. “Non-Abelian”1

Determinant

for 2x2 matrix

if \(A= \pmatrix {a & b \\ c & d}\), then \(det A = \|A\| = ad - be\)

for larger matrix

프로그래머를-위한-선형대수1-벡터,행렬,행렬식 포스트 참고

Inverse

\(AA^{-1} = I\), \(AC^T = C^TA = \|A\|I\)

\(\therefore I = \frac {AC^T} {\|A\|} = \frac {C^TA} {\|A\|}\) (if \(\|A\| \neq 0\))

Eigen value problem

  • \(A\|x\rangle = \lambda \|x\rangle\) : eigen value problem

  • \(\|x\rangle\): eigen vector
  • \(\lambda\): eigen value

\(A\|x_\lambda\rangle = \lambda \|x_\lambda\rangle\).

Hermitian matrix

A matrix \(A\) is the hermitian matrix if \((A^T)^* = A\) (\(^*\) = complex conjugate) 2

norm

\(\langle x_\lambda \| = (\|x_\lambda\rangle)^\dagger\), \(\langle x_\lambda \| x_\lambda \rangle\): inner product

\(\sqrt{ \langle x_\lambda \| x_\lambda \rangle}\) called norm of a vector \(\|x_\lambda\rangle\)

normalized vector

\(\|e_\lambda\rangle\) : normalized vector

\(\|e_\lambda \rangle = \frac 1 {\sqrt{ \langle x_\lambda \| x_\lambda \rangle}} \|x_\lambda \rangle\), \(\langle e_\lambda \|e_\lambda \rangle = \frac 1 { \langle x_\lambda \| x_\lambda \rangle} \langle x_\lambda \|x_\lambda \rangle = 1\)

Eigen value problem

  1. \(det (A - \lambda I) \neq 0\) : \(\|x_\lambda\rangle = 0\) : null vector
  2. \(det (A - \lambda I) = 0\) then we have an arbitrary solution.
  • we will be focusing on the “Hermitian matrix” only

Other matrices

Identity matrix

\(IA = AI = I\). Identity matrix I plays the same role as the number 1 in the multiplication of numbers.

Square matrix

  • square matrix: nxn matrix
  • closed to addition, subtraction and multiplication

Example : \(I = \pmatrix {1 & 0 & 0 & ... \\ 0 & 1 & 0 & ... \\ 0 & 0 & 1 & ... \\ ... & ... & ... & ... }\) 3

Column, Row Vector

\(\pmatrix {x_1 \\ x_2 \\ ...}\) : column vector (column matrix) or ket vector4

\(\pmatrix {x_1 & x_2 & ...}\) : row vector (row matrix) or bra vector4

braket notation

\(\langle x \| y \rangle = \pmatrix{x_1 & x_2 & ...} \pmatrix {y_1 \\ y_2 \\ ... }\).

다음에 더 정리해야지

Pauli matrix

Differential equation

Ordinary Differential Equation O.D.E.

Partial Differential Equation P.D.E.

Exact & Separable

O.D.E.

Integrating Factor

Homogeneous & Inhomogeneous

Equidimensional O.D.E.

Inhomogen

  1. https://en.wikipedia.org/wiki/Non-abelian_group 

  2. https://en.wikipedia.org/wiki/Conjugate_transpose 수업에서는 dagger로 주로 쓰셨다. 

  3. https://en.wikipedia.org/wiki/Kronecker_delta 

  4. https://en.wikipedia.org/wiki/Bra–ket_notation  2

August 22, 2018
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