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THE CS NOTES

Tutorial Mathematics (Gate Walaah) ▶️

Singular matrix -> Determinant of matrix = 0

If rank < min(m,n) -> singular matrix possible rank <=min(m,n)

Elementary operations

Echelon form -> settle all zeroes increasing order rowise -> to find rank

Matrics Derived from a Matrix using Elementary operations -> Equivalent matrix All 3 Elementeray transformation do not change rank -> Rank is same for equivalent Matrices -> Rank is invarient property of a matrix 3rd Elementary operation -> do not change determinant

Non homogeneous system

AX=B

  1. P(A)!=P(A:B) ->P(A)< min(no. of row) -> for one or more eques. LHS(A)=0 and RHS(B)=constant -> inconsistent solution
  2. P(A)=P(A:B) = no. of variable = P(B) -> for all eques LHS(A)=equs RHS(B)=constant -> Unique Solution
  3. P(A)=P(A:B) < no. of variable = P(B) -> A & B linearly Dependent -> for some eques LHS(A)=0 and RHS(0) -> Infinite Solution

|A|=0 -> Singular -> Linearly Dependent Colum -> No Unique Solution &

  1. (adj.A)B=0 -> infinite solution
  2. (adj.A)B!=0 -> no Solution
  3. |A|!=0 -> Non Linear Dependent Colum -> Unique Solution

B=0, Trivial solution = zero sollution = unique solution B!=0, non trivial solution = non - zero solution = infinite solution

Homogeneous system (AX=0) either (A=0 & B=0 infinite) or (A=const & B=0 unique) is always consistent,

Dolittle(left matrix diagonal unit) & Crout(right matrix diagonal unit) Method Lower upper triangle

Vector -> column Matrix

AX = Multiplication of Matrix A into Vector X λX = Multiplication of Number λ into Vector X

AX = λX Stretch Vector (if λ>1) or shrink vector if ( 0<λ<1) if ( λ<0 or negative) rotate of column vector by 180 and then shrink or stretch

λ : Special value that shrink or stretch vector λ no. of time

AX= λI X = λX AX- λX = (A- λI) = 0 λ = eigne value , X = eigen vector A- λ Characteristic matrix |A- λI| characteristic polynomial |A- λI| = 0 characteristic Equation

eigen vector is a column matrix

Eigen value of equivalent matrix need not be same, i.e. applying elmentary operation change eigen value.

Spectrum of A - set of eighen values of A Spectral Radius - Largest eigen value

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Factorisation - (x^3 - 9)/(x-3) at x = 3

0/0 or infinity/infinity ?? -> L hospital Rule

f(x)/g(x) = f(x)‘/g(x)’ = f(x)”/g(x)” = f(x)^n/g(x)^n

e^x in maclaurine series remember

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