Eigen Values and Eigen Vectors Flashcards
Equation used for eigen vector and eigen values
AX = λX
Give basic definition of eigen vector and eigen values
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.
Another name of Eigen value
Characteristic value of Aₙₓₙ
AX = λX
explain it in words
Matrix multiplication of vector = Scalar multiple of vector
Characteristic equation
|A - λI|=0
How to find eigen values
By solving characteristic equation i.e |A - λI|=0
How to find Eigen vectors
By solving (A - λI)X = 0. {here X≠0}
No. of Independent eigen vectors
(formula not definition)
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
General characteristic equation of A
|A-λI| = (-1)ⁿ λⁿ + (-1)ⁿ⁻¹ Trace(A) λⁿ⁻¹ + ——-+|Anxn| = 0
Aₙₓₙ has _______________ number of eigen values
Anxn has n number of eigen values
Note of General characteristic equation
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)
How do we get determinant of matrix by general characteristic equation
If coefficient of λⁿ is (-1)ⁿ then the constant term = determinant of matrix
How do we get Trace(A) by characteristic equation
If coefficient of λⁿ is (-1)ⁿ then
(-1)ⁿ⁻¹ Trace(A) = coefficient of λⁿ⁻¹
Sum of eigen values of matrix A
Trace(A)
Trace(A) in terms of eigen values
Trace(A) = Sum of eigen values of A
Product of eigen values
Determinant of matrix
i.e.|A|
Determinant of A in terms of eigen values
Product of eigen values
A and Aᵀ
explain in context of eigen values
eigen values of A = eigen values of Aᵀ
If |A| = 0 then
explain in context of eigen values
If |A| = 0 then at least one eigen value = 0
If |A|≠ 0
explain in context of eigen values
If |A|≠ 0 then none of the eigen value is 0
Eigen values of symmetric matrix
Real
Eigen values of skew-symmetric matrix
zero or purely imaginary
Eigen values of orthogonal matrix
|λ| = 1
i.e. Eigen values of orthogonal matrix are of unit modulus (|λ| = 1)
Eigen values of unitary matrix
|λ| = 1
i.e. Eigen values of unitary matrix are of unit modulus (|λ| = 1)
Rank of A
in context of eigen values
Number of non-zero eigen values
If λ is eigen value of A. Then
eigen value of Aᵏ __________
Eigen value of Aᵏ is λᵏ
If λ is eigen value of A. Then
eigen value of kA ___________
Eigen value of kA is kλ
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
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
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)
Eigen vectors of ____________________ are same
Eigen vectors of A, A², ……, Aᵏ ,A⁻¹, kA, adj A, polynomial matrix in A, eᴬ are same
Eigen values of Involuntary matrix
are ±1
Eigen values of Idempotent matrix
are 0,1
Eigen values of Lower Triangular matrix, Upper Triangular matrix
are principal diagonal elements
Eigen values of Diagonal matrix
are principal diagonal elements
Definition of eigen value in terms of matrix
Every matrix can be replaced by its eigen value
Number of eigen values
Number of eigen values = Order of matrix i.e. n
Eigen values of Nilpotent Matrix
Eigen values of Nilpotent matrix are zero
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
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)|
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)
Eigen values of AAᵀ
Eigen values of AAᵀ are always greater or equal to zero. i.e. λ(AAᵀ) ≥ 0
Eigen Values of AAᵀ are real or not
give reason also
eigen values of AAᵀ are always real because AAᵀ is symmetric matrix
Eigen values of AdjA if λ is eigen value of Non-singular matrix
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]
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
Row sum or column sum shortcut in eigen values
Diagonal element and eigen value shorcut
Uniqueness of eigen vector property
Eigen vector corresponding to an eigen value is not uniques (infinitely many)
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.
When matrix can’t be diagonalized
When number of independent eigen vectors is not equal to order of matrix
Eigen value and number of independent vector relation
Non-repeated eigen value will have exactly one independent eigen vector.
When will eigen values positive (property)
Eigen values of Anxn are real and positive if all the principal minors are positive
Cayley Hamilton theorem
Characteristic equation of A3x3
