p.1
Definition of Eigenvectors
What are eigenvectors?
Exceptional vectors that remain in the same direction as Ax.
p.5
Examples of Eigenvalues and Eigenvectors
What is the main question regarding vectors u and v?
Are u and v eigenvectors of the matrix A?
p.3
Definition of Eigenvectors
What is an eigenvector of an n × n matrix?
A nonzero vector 𝐱 such that 𝐀𝐱 = 𝛌 𝐱 for some scalar 𝛌.
What does the equation Ax = 5 yield for λ2 = 5?
An eigenvector y z = 1 2.
p.9
Properties of Eigenvalues and Eigenvectors
What condition must be satisfied for B to have no inverse in relation to eigenvalues?
det(B) = det(A - λI) = 0.
p.7
Properties of Eigenvalues and Eigenvectors
What condition indicates that (A - λI) is not invertible?
If (A - λI)x = 0 has a nonzero solution.
p.7
Characteristic Equation
What must the determinant of (A - λI) be for eigenvalues?
The determinant must be 0.
What does the equation Ax = 0 yield for λ1 = 0?
An eigenvector y z = 2 - 1.
p.9
Characteristic Equation
What does the equation (A - λI)x = 0 represent?
It represents the characteristic equation for finding eigenvalues.
What is the significance of the equation A - λI x = 0?
It is used to solve for the eigenvector x corresponding to the eigenvalue λ.
p.9
Properties of Eigenvalues and Eigenvectors
Can the zero vector be an eigenvector?
No, the zero vector cannot be an eigenvector.
p.2
Properties of Eigenvalues and Eigenvectors
What does the result Av indicate about the vector v?
Av is just 2v, meaning A stretches or dilates the vector v.
p.8
Definition of Eigenvalues
What is an eigenvalue of a matrix A?
The number λ is an eigenvalue of A if and only if A − λI is singular.
p.4
Definition of Eigenvectors
What is the equation representing the relationship between a matrix and its eigenvector?
𝐀𝐱 = 𝜆𝐱, where 𝐀 is the matrix, 𝜆 is the eigenvalue, and 𝐱 is the eigenvector.
p.4
Definition of Eigenvalues
What does 𝜆 represent in the context of eigenvalues?
𝜆 represents the eigenvalue (also known as proper values, characteristic values, or latent roots).
What does each eigenvalue λ lead to?
Each λ leads to an eigenvector.
p.3
Definition of Eigenvalues
What is an eigenvalue?
A scalar 𝛌 is called an eigenvalue of A if there is a nontrivial solution 𝐱 of 𝐀𝐱 = 𝛌 𝐱.
Once eigenvalues are found, what is the next step?
Find the corresponding eigenvectors.
What does the equation Ax = λx represent?
It represents the relationship between a matrix A, an eigenvalue λ, and its corresponding eigenvector x.
p.9
Definition of Eigenvectors
What is the equation that defines an eigenvector and eigenvalue?
Ax = λx, where A is a square matrix, x is a nonzero vector, and λ is the eigenvalue.
p.11
Characteristic Equation
What is the first step to find the eigenvalues of matrix A?
Calculate the characteristic polynomial.
p.1
Examples of Eigenvalues and Eigenvectors
Given A = [[3, -2], [1, 0]], u = [-1, 1], and v = [2, 1], what do you need to do to visualize the eigenvectors?
Draw the vectors Au and Av.
p.3
Definition of Eigenvectors
What is the relationship between eigenvalues and eigenvectors?
An eigenvector corresponds to an eigenvalue, satisfying the equation 𝐀𝐱 = 𝛌 𝐱.
p.5
Examples of Eigenvalues and Eigenvectors
What are the vectors u and v in Example 2?
u = [6, -5] and v = [3, -2].
How many eigenvalues does an n × n matrix A have?
A has n eigenvalues, which can be repeated.
p.1
Definition of Eigenvectors
What is the relationship between an eigenvector x and its transformation Ax?
The vector Ax is a number λ times the original vector x.
p.4
Definition of Eigenvalues
In the equation 𝐀𝐱 = 𝜆𝐱, what does 𝐀 represent?
𝐀 represents the matrix associated with the eigenvalue problem.
p.2
Properties of Eigenvalues and Eigenvectors
What does the operation A do to the vector v?
A only stretches or dilates the vector v.
What is the expression for A - λI?
A - λI = [[1 - λ, 2], [2, 4 - λ]].
p.4
Definition of Eigenvectors
What is the general form of a matrix equation involving eigenvalues and eigenvectors?
The general form is 𝐀𝐱 = 𝜆𝐱.
p.9
Properties of Eigenvalues and Eigenvectors
What happens if B has an inverse and Bx = 0?
If B has an inverse, then Bx = 0 implies x = 0, which contradicts the definition of an eigenvector.
How do you rewrite the equation Ax = λx to find eigenvalues?
As Ax = λIx ⇔ Ax - λIx = 0 ⇔ (A - λI)x = 0.
p.10
Definition of Eigenvectors
What does the term 'eigenvector' refer to?
A non-zero vector that changes only by a scalar factor when a linear transformation is applied.
Is v an eigenvector of A?
No, because Av is not a multiple of v.
p.9
Definition of Eigenvalues
What does λ represent in the context of eigenvectors?
λ represents the eigenvalue associated with the eigenvector x.
What do you obtain after solving the characteristic equation for matrix A?
The eigenvalues of the matrix.
What is the first step to solve the eigenvalue and eigenvector problem for an n × n matrix?
Compute the determinant of A - λI, which will be a polynomial in λ with degree n.
What condition do the eigenvalues satisfy regarding the matrix A?
They make A - λI singular.
p.4
Matrix Multiplication and Eigenvectors
What does the equation 𝐀𝐱 = 𝐛 represent?
It represents a system of linear equations.
p.3
Properties of Eigenvalues and Eigenvectors
What happens when 𝐀 multiplies an eigenvector 𝐱?
It dilates, contracts, or reverses the direction of 𝐱, depending on the value of 𝛌.
p.6
Properties of Eigenvalues and Eigenvectors
What does the equation Au = -4u signify?
It shows that u is an eigenvector of A with eigenvalue -4.
