Eigen Values and Eigen Vectors

Question 1

Equation used for eigen vector and eigen values

Answer 1

AX = λX

Question 2

Give basic definition of eigen vector and eigen values

Answer 2

Let Aₙₓₙ for any scalar λ, ∃ X≠0 (non-zero vector) such that AX = λX
λ is called eigen value of Anxn
and
X ≠ 0 is called eigen vector corresponding to λ.

Question 3

Another name of Eigen value

Answer 3

Characteristic value of Aₙₓₙ

Question 4

AX = λX
explain it in words

Answer 4

Matrix multiplication of vector = Scalar multiple of vector

Question 5

Characteristic equation

Answer 5

|A - λI|=0

Question 6

How to find eigen values

Answer 6

By solving characteristic equation i.e |A - λI|=0

Question 7

How to find Eigen vectors

Answer 7

By solving (A - λI)X = 0. {here X≠0}

Question 8

No. of Independent eigen vectors

(formula not definition)

Answer 8

No. of independent eigen vectors for an eigen value λ = n - rank(A-λI) = no. of free variable of A-λI
=geometric multiplicity of λ

Question 9

General characteristic equation of A

Answer 9

|A-λI| = (-1)ⁿ λⁿ + (-1)ⁿ⁻¹ Trace(A) λⁿ⁻¹ + ——-+|Anxn| = 0

Question 10

Aₙₓₙ has _______________ number of eigen values

Answer 10

Anxn has n number of eigen values

Question 11

Note of General characteristic equation

Answer 11

If coefficient of λⁿ is (-1)ⁿ then constant term of characteristic polynomial of Aₙₓₙ = |A| = determinant of A
and
coefficient of λⁿ⁻¹ = (-1)ⁿ⁻¹ Trace(A)

Question 12

How do we get determinant of matrix by general characteristic equation

Answer 12

If coefficient of λⁿ is (-1)ⁿ then the constant term = determinant of matrix

Question 13

How do we get Trace(A) by characteristic equation

Answer 13

If coefficient of λⁿ is (-1)ⁿ then
(-1)ⁿ⁻¹ Trace(A) = coefficient of λⁿ⁻¹

Question 14

Sum of eigen values of matrix A

Answer 14

Trace(A)

Question 15

Trace(A) in terms of eigen values

Answer 15

Trace(A) = Sum of eigen values of A

Question 16

Product of eigen values

Answer 16

Determinant of matrix
i.e.|A|

Question 17

Determinant of A in terms of eigen values

Answer 17

Product of eigen values

Question 18

A and Aᵀ
explain in context of eigen values

Answer 18

eigen values of A = eigen values of Aᵀ

Question 19

If |A| = 0 then
explain in context of eigen values

Answer 19

If |A| = 0 then at least one eigen value = 0

Question 20

If |A|≠ 0
explain in context of eigen values

Answer 20

If |A|≠ 0 then none of the eigen value is 0

Question 21

Eigen values of symmetric matrix

Answer 21

Real

Question 22

Eigen values of skew-symmetric matrix

Answer 22

zero or purely imaginary

Question 23

Eigen values of orthogonal matrix

Answer 23

|λ| = 1
i.e. Eigen values of orthogonal matrix are of unit modulus (|λ| = 1)

Question 24

Eigen values of unitary matrix

Answer 24

|λ| = 1
i.e. Eigen values of unitary matrix are of unit modulus (|λ| = 1)

Question 25

Rank of A
in context of eigen values

Answer 25

Number of non-zero eigen values

Question 26

If λ is eigen value of A. Then
eigen value of Aᵏ __________

Answer 26

Eigen value of Aᵏ is λᵏ

Question 27

If λ is eigen value of A. Then
eigen value of kA ___________

Answer 27

Eigen value of kA is kλ

Question 28

If λ is eigen value of A. The eigen value of aₙ Aⁿ +aₙ₋₁ Aⁿ⁻¹+——+ a₁A + a₀I is

Answer 28

Eigen value of aₙ Aⁿ +aₙ₋₁ Aⁿ⁻¹+——+ a₁A + a₀I is
aₙ λⁿ +aₙ₋₁ λⁿ⁻¹+——+ a₁λ+a₀

Question 29

Special about A, A², ……, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ

Answer 29

Eigen vectors of A, A², ……, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ are same

Question 30

If λ is eigen value of ______________ then
1.) _________ is eigen value of A⁻¹
2.) _________ is eigen value of (adj A)

Answer 30

If λ is eigen value of non-singular matrix then
1.) 1/λ is eigen value of A⁻¹
2.) |A|/λ is eigen value of (adj A)

Question 31

Eigen vectors of ____________________ are same

Answer 31

Eigen vectors of A, A², ……, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ are same

Question 32

Eigen values of Involuntary matrix

Answer 32

are ±1

Question 33

Eigen values of Idempotent matrix

Answer 33

are 0,1

Question 34

Eigen values of Lower Triangular matrix, Upper Triangular matrix

Answer 34

are principal diagonal elements

Question 35

Eigen values of Diagonal matrix

Answer 35

are principal diagonal elements

Question 36

Definition of eigen value in terms of matrix

Answer 36

Every matrix can be replaced by its eigen value

Question 37

Number of eigen values

Answer 37

Number of eigen values = Order of matrix i.e. n

Question 38

Eigen values of Nilpotent Matrix

Answer 38

Eigen values of Nilpotent matrix are zero

Question 39

If one eigen value of real matrix is complex then what about its other eigen value

Answer 39

The eigen values of real matrix must occur in complex conjugate pair if any of the eigen values are complex

Question 40

Absolute value of the product eigen values

Answer 40

Absolute value of the product eigen values = absolute value of determinant of matrix
|λ₁ λ₂ λ₃ ……… λₙ | = |det(A)|

Question 41

Eigen values of Identity matrix of order nxn

Answer 41

The eigenvalues of an identity matrix I of order nxn are all** 1**
In other words, the eigenvalues of Iₙ are 1,1,1,….,1(repeated n times)

Question 42

Eigen values of AAᵀ

Answer 42

Eigen values of AAᵀ are always greater or equal to zero. i.e. λ(AAᵀ) ≥ 0

Question 43

Eigen Values of AAᵀ are real or not
give reason also

Answer 43

eigen values of AAᵀ are always real because AAᵀ is symmetric matrix

Question 44

Eigen values of AdjA if λ is eigen value of Non-singular matrix

Answer 44

Question 45

Non-zero eigenvalues of AAᵀ is

Answer 45

Number of non-zero eigen values of AAᵀ is equal to rank A
since [rank A = rank of AAᵀ]

Question 46

Compare the geometric and algebraic multiplicity of λ

Answer 46

Geometric multiplicity of λ ≤ Algebraic multiplicity of λ
i.e. The geometric of an eigen value λ of a Matrix A doesn’t exceed its algebraic multiplicity