Midterm 3 Flashcards
What is an eigenvalue?
An eigenvalue is a scalar that indicates how much an eigenvector is stretched or compressed during a linear transformation.
True or False: Eigenvectors can only be found for square matrices.
True
Fill in the blank: The equation Ax = λx represents the relationship between a matrix A, an eigenvalue λ, and an eigenvector x.
λ
What is the characteristic polynomial used for?
It is used to find the eigenvalues of a matrix by setting the determinant of (A - λI) to zero.
What does the symbol λ represent in the context of eigenvalues?
λ represents an eigenvalue.
What is the geometric interpretation of an eigenvector?
An eigenvector points in a direction that remains unchanged by the transformation represented by the matrix.
How do you calculate eigenvalues for a 2x2 matrix?
By solving the characteristic equation det(A - λI) = 0.
Multiple Choice: Which of the following is true about eigenvalues? A) They can be negative B) They must be real C) They are always positive D) None of the above
A) They can be negative
What is an eigenvector?
An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied.
True or False: Eigenvalues can be complex numbers.
True
What does the matrix A represent in the equation Ax = λx?
A represents a linear transformation or a square matrix.
Fill in the blank: The set of all eigenvalues of a matrix is called its _______.
spectrum
What is the algebraic multiplicity of an eigenvalue?
It is the number of times an eigenvalue appears as a root of the characteristic polynomial.
What is the geometric multiplicity of an eigenvalue?
It is the number of linearly independent eigenvectors associated with that eigenvalue.
Multiple Choice: Which of the following methods can be used to find eigenvalues? A) Power method B) QR algorithm C) Both A and B D) None of the above
C) Both A and B
What is a diagonalizable matrix?
A matrix is diagonalizable if it can be expressed in the form PDP⁻¹, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors.
True or False: All matrices have real eigenvalues.
False
What condition must be met for a matrix to have a complete set of eigenvectors?
The matrix must be diagonalizable.
How is the eigenvalue problem generally expressed?
It is expressed as Ax = λx, where A is a matrix, λ is an eigenvalue, and x is an eigenvector.
What is the significance of eigenvalues in systems of differential equations?
They determine the stability and behavior of the system’s solutions.
Fill in the blank: The eigenvalues of a symmetric matrix are always _______.
real
What is the relationship between the eigenvalues of a matrix and its determinant?
The determinant of a matrix is the product of its eigenvalues.
Multiple Choice: Which of the following is NOT a property of eigenvalues? A) They can be complex B) They can be negative C) They must be integers D) They can be repeated
C) They must be integers
What does it mean for an eigenvalue to have an algebraic multiplicity greater than its geometric multiplicity?
It indicates that there are fewer linearly independent eigenvectors than the number of times the eigenvalue occurs.
