Diagonalization, Quadratic forms Flashcards
Explain Diagonalization
In diagonalization, which transformation used?
Similarity transformation
Aₙₓₙ can be diagonalised using similarity transformation B=P⁻¹AP where P is non-singular matrix
In P⁻¹AP = D
What is P here
P is non-singular matrix which is given by P = [X₁ X₂ X₃ …….Xₙ]ₙₓₙ consisting of all n independent eigen vectors of Aₙₓₙ
P is called Modal matrix
Explain similarity transformation
P⁻¹AP = B
Here A and B are similar matrices i.e. eigen values of A and B are same
P⁻¹AP = B
Tell us about A and B here
Here A and B are similar matrices i.e.
eigen values of A and B are same
Necessary and sufficient condition for Aₙₓₙ to get diagonalised
no. of independent eigen vectors of Aₙₓₙ = order of Aₙₓₙ = n
D =P⁻¹AP
what is D here
If A and D are similar matrices what about their determinants
|A|=|D|
________________ is also very important in getting diagonalized matrix
Order of eigen values is also very important in getting diagonalized matrix
Explain Dⁿ if D is given
If D = P⁻¹AP then Dⁿ = P⁻¹AⁿP
Explain A if D is given
If D = P⁻¹AP then A = PDP⁻¹
Explain Aⁿ if A is given
If A = PDP⁻¹ then Aⁿ = PDⁿP⁻¹
Explain eᴰ if D is given
If D = P⁻¹AP then eᴰ = P⁻¹eᴬP
Explain eᴬ if D is given
If D = P⁻¹AP then eᴬ = PeᴰP⁻¹
State transition matrix
If D = P⁻¹AP then eᴬ = PeᴰP⁻¹
here eᴬ is transition matrix
Explain Quadratic Form
What type of matrix is in Quadratic From
Real symmetric matrix
Quadratic Form only definition
Xₙₓ₁ vector and Aₙₓₙ is a real symmetric matrix then XᵀAX is called quadratic form
Equation of Quadratic form
Special type of quadratic form
Orthogonal Form
Another names of orthogonal form (2)
Canonical Form or Sum of squares of quadratic form
Explain orthogonal form
Orthogonal form of quadratic form (only definition)
It is given by YᵀDY where Y = [Y₁Y₂….Yₙ]ᵀ, Y∈Rⁿ
Equation of orthogonal form
Orthogonal form of quadratic form: λ₁Y₁²+ λ₂Y₂²+ λ₃Y₃²+ …..+ λₙYₙ²
Nature of quadratic form (5)
1.) If XᵀAX > 0 ∀ X∈Rⁿ
OR
λ>0,
then Q.F. is positive definite
2.) If XᵀAX ≥ 0 ∀ X∈Rⁿ
OR
λ≥0,
then Q.F. is semi positive definite
3.) If XᵀAX < 0 ∀ X∈Rⁿ
OR
λ<0,
then Q.F. is negative definite
4.) If XᵀAX ≤ 0 ∀ X∈Rⁿ
OR
λ≤0,
then Q.F. is semi negative definite
5.) If XᵀAX>0 and ≤ 0 ∀ X∈Rⁿ
OR
λ>0 and ≤0,
then Q.F. is indefinite
(Here λ is eigen value of A)
Condition for finding nature of quadratic form
In XᵀAX, X≠0
Q.F. is positive definite
XᵀAX > 0 ∀ X∈Rⁿ
OR
λ>0
Q.F. is semi positive definite
XᵀAX ≥ 0 ∀ X∈Rⁿ
OR
λ≥0
Q.F. is negative definite
XᵀAX < 0 ∀ X∈Rⁿ
OR
λ<0
Q.F. is semi-negative definite
XᵀAX ≤ 0 ∀ X∈Rⁿ
OR
λ≤0
Q.F is indefinite
If XᵀAX>0 and ≤ 0 ∀ X∈Rⁿ
OR
λ>0 and ≤0
XᵀAX > 0 ∀ X∈Rⁿ
Tell the nature of Q.F
Q.F. is positive definite
λ>0
Tell the nature of Q.F.
Q.F. is positive definite
XᵀAX ≥ 0 ∀ X∈Rⁿ
Tell the nature of Q.F.
Q.F. is semi positive definite
λ≥0
Tell the nature of Q.F.
Q.F. is semi positive definite
XᵀAX < 0 ∀ X∈Rⁿ
Tell the nature of Q.F.
Q.F. is negative definite
λ<0
Tell the nature of Q.F.
Q.F. is negative definite
XᵀAX ≤ 0 ∀ X∈Rⁿ
Tell the nature of Q.F.
Q.F. is semi-negative definite
λ≤0
Tell the nature of Q.F.
Q.F. is semi-negative definite
If XᵀAX>0 and ≤ 0 ∀ X∈Rⁿ
Tell the nature of Q.F.
Q.F is indefinite
λ>0 and ≤0
Tell the nature of Q.F.
Q.F is indefinite
Rank of Quadratic form
Rank of Q.F. = Rank of A = no. of non-zero eigen values of A = r
How to denote
Index of Q.F
p
Index of Q.F
p = number of positive eigen values of A
What is Signature of Q.F.
excess +ve compared to -ve λ of A
Formula of signature of Q.F.
2p - r
Here p is index of Q.F. and r is rank of Q.F.
