Diagonalization, Quadratic forms

Question 1

Explain Diagonalization

Answer 1

Question 2

In diagonalization, which transformation used?

Answer 2

Similarity transformation
Aₙₓₙ can be diagonalised using similarity transformation B=P⁻¹AP where P is non-singular matrix

Question 3

In P⁻¹AP = D
What is P here

Answer 3

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

Question 4

Explain similarity transformation

Answer 4

P⁻¹AP = B
Here A and B are similar matrices i.e. eigen values of A and B are same

Question 5

P⁻¹AP = B
Tell us about A and B here

Answer 5

Here A and B are similar matrices i.e.
eigen values of A and B are same

Question 6

Necessary and sufficient condition for Aₙₓₙ to get diagonalised

Answer 6

no. of independent eigen vectors of Aₙₓₙ = order of Aₙₓₙ = n

Question 7

D =P⁻¹AP
what is D here

Answer 7

Question 8

If A and D are similar matrices what about their determinants

Answer 8

|A|=|D|

Question 9

________________ is also very important in getting diagonalized matrix

Answer 9

Order of eigen values is also very important in getting diagonalized matrix

Question 10

Explain Dⁿ if D is given

Answer 10

If D = P⁻¹AP then Dⁿ = P⁻¹AⁿP

Question 11

Explain A if D is given

Answer 11

If D = P⁻¹AP then A = PDP⁻¹

Question 12

Explain Aⁿ if A is given

Answer 12

If A = PDP⁻¹ then Aⁿ = PDⁿP⁻¹

Question 13

Explain eᴰ if D is given

Answer 13

If D = P⁻¹AP then eᴰ = P⁻¹eᴬP

Question 14

Explain eᴬ if D is given

Answer 14

If D = P⁻¹AP then eᴬ = PeᴰP⁻¹

Question 15

State transition matrix

Answer 15

If D = P⁻¹AP then eᴬ = PeᴰP⁻¹
here eᴬ is transition matrix

Question 16

Explain Quadratic Form

Answer 16

Question 17

What type of matrix is in Quadratic From

Answer 17

Real symmetric matrix

Question 18

Quadratic Form only definition

Answer 18

Xₙₓ₁ vector and Aₙₓₙ is a real symmetric matrix then XᵀAX is called quadratic form

Question 19

Equation of Quadratic form

Answer 19

Question 20

Special type of quadratic form

Answer 20

Orthogonal Form

Question 21

Another names of orthogonal form (2)

Answer 21

Canonical Form or Sum of squares of quadratic form

Question 22

Explain orthogonal form

Answer 22

Question 23

Orthogonal form of quadratic form (only definition)

Answer 23

It is given by YᵀDY where Y = [Y₁Y₂….Yₙ]ᵀ, Y∈Rⁿ

Question 24

Equation of orthogonal form

Answer 24

Orthogonal form of quadratic form: λ₁Y₁²+ λ₂Y₂²+ λ₃Y₃²+ …..+ λₙYₙ²

Question 25

Nature of quadratic form (5)

Answer 25

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)

Question 26

Condition for finding nature of quadratic form

Answer 26

In XᵀAX, X≠0

Question 27

Q.F. is positive definite

Answer 27

XᵀAX > 0 ∀ X∈Rⁿ OR λ>0

Question 28

Q.F. is semi positive definite

Answer 28

XᵀAX ≥ 0 ∀ X∈Rⁿ OR λ≥0

Question 29

Q.F. is negative definite

Answer 29

XᵀAX < 0 ∀ X∈Rⁿ OR λ<0

Question 30

Q.F. is semi-negative definite

Answer 30

XᵀAX ≤ 0 ∀ X∈Rⁿ OR λ≤0

Question 31

Q.F is indefinite

Answer 31

If XᵀAX>0 and ≤ 0 ∀ X∈Rⁿ OR λ>0 and ≤0

Question 32

XᵀAX > 0 ∀ X∈Rⁿ Tell the nature of Q.F

Answer 32

Q.F. is positive definite

Question 33

λ>0 Tell the nature of Q.F.

Answer 33

Q.F. is positive definite

Question 34

XᵀAX ≥ 0 ∀ X∈Rⁿ Tell the nature of Q.F.

Answer 34

Q.F. is semi positive definite

Question 35

λ≥0 Tell the nature of Q.F.

Answer 35

Q.F. is semi positive definite

Question 36

XᵀAX < 0 ∀ X∈Rⁿ Tell the nature of Q.F.

Answer 36

Q.F. is negative definite

Question 37

λ<0 Tell the nature of Q.F.

Answer 37

Q.F. is negative definite

Question 38

XᵀAX ≤ 0 ∀ X∈Rⁿ Tell the nature of Q.F.

Answer 38

Q.F. is semi-negative definite

Question 39

λ≤0 Tell the nature of Q.F.

Answer 39

Q.F. is semi-negative definite

Question 40

If XᵀAX>0 and ≤ 0 ∀ X∈Rⁿ Tell the nature of Q.F.

Answer 40

Q.F is indefinite

Question 41

λ>0 and ≤0 Tell the nature of Q.F.

Answer 41

Q.F is indefinite

Question 42

Rank of Quadratic form

Answer 42

Rank of Q.F. = Rank of A = no. of non-zero eigen values of A = r

Question 43

How to denote Index of Q.F

Answer 43

p

Question 44

Index of Q.F

Answer 44

p = number of positive eigen values of A

Question 45

What is Signature of Q.F.

Answer 45

excess +ve compared to -ve λ of A

Question 46

Formula of signature of Q.F.

Answer 46

2p - r Here p is index of Q.F. and r is rank of Q.F.