Eigen Values and Eigen Vectors Flashcards

1
Q

Equation used for eigen vector and eigen values

A

AX = λX

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2
Q

Give basic definition of eigen vector and eigen values

A

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.

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3
Q

Another name of Eigen value

A

Characteristic value of Aₙₓₙ

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4
Q

AX = λX
explain it in words

A

Matrix multiplication of vector = Scalar multiple of vector

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5
Q

Characteristic equation

A

|A - λI|=0

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6
Q

How to find eigen values

A

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

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7
Q

How to find Eigen vectors

A

By solving (A - λI)X = 0. {here X≠0}

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8
Q

No. of Independent eigen vectors

(formula not definition)

A

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

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9
Q

General characteristic equation of A

A

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

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10
Q

Aₙₓₙ has _______________ number of eigen values

A

Anxn has n number of eigen values

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11
Q

Note of General characteristic equation

A

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)

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12
Q

How do we get determinant of matrix by general characteristic equation

A

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

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13
Q

How do we get Trace(A) by characteristic equation

A

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

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14
Q

Sum of eigen values of matrix A

A

Trace(A)

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15
Q

Trace(A) in terms of eigen values

A

Trace(A) = Sum of eigen values of A

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16
Q

Product of eigen values

A

Determinant of matrix
i.e.|A|

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17
Q

Determinant of A in terms of eigen values

A

Product of eigen values

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18
Q

A and Aᵀ
explain in context of eigen values

A

eigen values of A = eigen values of Aᵀ

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19
Q

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

A

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

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20
Q

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

A

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

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21
Q

Eigen values of symmetric matrix

A

Real

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22
Q

Eigen values of skew-symmetric matrix

A

zero or purely imaginary

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23
Q

Eigen values of orthogonal matrix

A

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

24
Q

Eigen values of unitary matrix

A

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

25
Rank of A in context of eigen values
Number of non-zero eigen values
26
If λ is eigen value of A. Then eigen value of Aᵏ __________
Eigen value of Aᵏ is λᵏ
27
If λ is eigen value of A. Then eigen value of kA ___________
Eigen value of kA is kλ
28
If λ is eigen value of A. The eigen value of aₙ Aⁿ +aₙ₋₁ Aⁿ⁻¹+------+ a₁A + a₀I is
Eigen value of aₙ Aⁿ +aₙ₋₁ Aⁿ⁻¹+------+ a₁A + a₀I is **aₙ λⁿ +aₙ₋₁ λⁿ⁻¹+------+ a₁λ+a₀**
29
Special about A, A², ......, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ
Eigen vectors of A, A², ......, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ are same
30
If λ is eigen value of ______________ then 1.) _________ is eigen value of A⁻¹ 2.) _________ is eigen value of (adj A)
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)
31
Eigen vectors of ____________________ are same
Eigen vectors of **A, A², ......, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ** are same
32
Eigen values of Involuntary matrix
are ±1
33
Eigen values of Idempotent matrix
are 0,1
34
Eigen values of Lower Triangular matrix, Upper Triangular matrix
are principal diagonal elements
35
Eigen values of Diagonal matrix
are principal diagonal elements
36
Definition of eigen value in terms of matrix
Every matrix can be replaced by its eigen value
37
Number of eigen values
Number of eigen values = Order of matrix i.e. n
38
Eigen values of Nilpotent Matrix
Eigen values of Nilpotent matrix are **zero**
39
If one eigen value of real matrix is complex then what about its other eigen value
The eigen values of real matrix must occur in **complex conjugate pair** if any of the eigen values are complex
40
Absolute value of the product eigen values
Absolute value of the product eigen values = absolute value of determinant of matrix |λ₁ λ₂ λ₃ ......... λₙ | = |det(A)|
41
Eigen values of Identity matrix of order nxn
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)
42
Eigen values of AAᵀ
Eigen values of AAᵀ are always greater or equal to zero. i.e. λ(AAᵀ) ≥ 0
43
Eigen Values of AAᵀ are real or not give reason also
eigen values of AAᵀ are always real because AAᵀ is symmetric matrix
44
Eigen values of AdjA if λ is eigen value of Non-singular matrix
45
Non-zero eigenvalues of AAᵀ is
Number of non-zero eigen values of AAᵀ is equal to rank A since [rank A = rank of AAᵀ]
46
Compare the geometric and algebraic multiplicity of λ
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
47
Row sum or column sum shortcut in eigen values
48
Diagonal element and eigen value shorcut
49
Uniqueness of eigen vector property
Eigen vector corresponding to an eigen value is not uniques (infinitely many)
50
Whose matrices eigen vectors are orthogonal
The eigen vectors corresponding to distinct eigen values of real symmetric matrix, hermitian matrix and orthogonal matrix are orthogonal to each other.
51
When matrix can't be diagonalized
When number of independent eigen vectors is not equal to order of matrix
52
Eigen value and number of independent vector relation
Non-repeated eigen value will have exactly one independent eigen vector.
53
When will eigen values positive (property)
Eigen values of Anxn are real and positive if all the principal minors are positive
54
Cayley Hamilton theorem
55
Characteristic equation of A3x3