The matrix has one eigenvalue equal to 3. The sum of the other two eigenvalues is
Sum of the eigen values of matrix is = Sum of diagonal values present in the matrix
∴ 1 + 0 + P = 3 + λ_{2 }+ λ_{3}
⇒ P + 1 = 3 + λ_{2 }+ λ_{3}
⇒ λ_{2 }+ λ_{3 }= P + 1 – 3 = P – 2
What is the determinant of matrix X if 4 and (2 + 7i) are the eigenvalues of X where i = √−1?
Two eigen value of X is 4 and (2 + 7i)
∴ (2  7i) (conjugate roots) must be the third root
Determinant of P = product of eigenvalues
Δ = 4 × (2 + 7i) × (2  7i)
Δ = 212
In the given matrix one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
For a given matrix A if V is the eigen vector corresponding to the eigen value λ, then:
AV = λV
∴ {α (−4, 2, 1) α ≠ 0, αϵR} are the corresponding eigenvectors.
Which of the belowgiven statements is/are true?
I. The eigenvalue of the lower triangular matrix is just the diagonal elements of the matrix.
II. The product of the eigenvalue of a matrix is equal to its trace.
III. If 1/λ is an eigenvalue of A’(inverse of A) then orthogonal of A also have 1/λ as its eigenvalue.
Example:
Eigenvalues are 1, 4 and 6 (diagonal elements)
Product of eigen value = determinants = 1 × 4 × 6 = 24
Orthogonal matrix and Inverse of given matrix have eigenvalues: 1,1/4 and 1/6
Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are  1 and 7. What are the values of a and b?
W.K.T.∑λ_{i }= ∑a_{ii}
∴  1 + 7 = 1 + a ⇒ a = 5
Also π λ_{i} = A
∴  1 × 7 = a – 4b
 7 = 5 – 4b ⇒ b = 3
Let A be the 2 X 2 matrix with elements a_{11} = 2, a_{12} = 3, a_{21} = 1 and a_{22} = 4 then the eigenvalues of the Matrix are A^{5}?
Given Matrix:
Characteristic Equation:
(2−λ) (4−λ) −3 = 0
λ2 − 6λ + 5 = 0
λ = 5 or λ = 1
The latent values of the matrix
are 1, 1, 2 and the number of linearly independent latent vectors for the repeated root 1 is –
Eigen vectors are also called invariant vectors, characteristic vectors, or latent vectors. Eigenvalues are also called characteristic roots or latent roots.
(A  λI)X = 0
Given λ = 1
⇒ (A  I) X = 0
⇒ x = 2y + 3x = 0, 10x – 5y + 5z = 0, 5x – 4y + 5z = 0
By solving above equations, we get x =
i.e. one linearly independent latent vector.
Consider a Matrix M = u^{T}v^{T} where u = (112) and v = also u^{T} denotes the transpose of matrix u. Find the largest eigenvalue of M?
Characteristic equation is given
λ^{3} − 5λ^{2} = 0
λ = 0 or λ = 5
Therefore largest value is 5.
What is the absolute difference of the eigenvalues for the matrix ad – bc = 6 and a + d = 7?
Let λ1 and λ2 be the two eigen values
Product of eigen value is equal to determinant of matrix
Sum of eigen value is equal to its trace
a + d = λ1 + λ2 = 7
λ1 × (7 − λ1) = 6
λ1^{2} − 7λ + 6 = 0
λ1 = 6 or λ2 = 1
λ1 − λ2 = 5
Consider the matrix which one of the following statements is TRUE for the eigenvalues and eigenvectors of the matrix?
(5  λ) (1  λ) + 4 = 0
5 – 5λ  λ + λ^{2} + 4 = 0
λ^{2}  6λ + 9 = 0
(λ  3)^{2} = 0
λ = 3
for eigen vector
So, only one independent eigen vector.
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