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Questions and Answers
What is the characteristic polynomial of matrix A if A = [1 2; 3 4]?
What is the characteristic polynomial of matrix A if A = [1 2; 3 4]?
- λ² - 5λ + 6
- λ² + 2λ - 8
- λ² - 4λ - 5 (correct)
- λ² - 3λ - 4
Which of the following expressions correctly represents the characteristic equation derived from the characteristic polynomial λ² - 4λ - 5?
Which of the following expressions correctly represents the characteristic equation derived from the characteristic polynomial λ² - 4λ - 5?
- λ² - 4λ + 5 = 0
- λ² + 4λ - 5 = 0
- λ² - 4λ - 5 = 0 (correct)
- λ² + 4λ + 5 = 0
If matrix A has eigenvalues λ = 5, -3, 7/3, what is the sum of the eigenvalues?
If matrix A has eigenvalues λ = 5, -3, 7/3, what is the sum of the eigenvalues?
- -1 (correct)
- 3
- 10
- 5
For the matrix B = 5A⁴ - 7A³ + 2A - 3A¹, what is the method to determine the eigenvalues of B based on the eigenvalues of A?
For the matrix B = 5A⁴ - 7A³ + 2A - 3A¹, what is the method to determine the eigenvalues of B based on the eigenvalues of A?
What is the determinant of the matrix A - λI if A = [2 0; 2 2]?
What is the determinant of the matrix A - λI if A = [2 0; 2 2]?
What is the characteristic equation for a 3x3 matrix A?
What is the characteristic equation for a 3x3 matrix A?
If the eigenvalues of matrix A are all equal to 2, which of the following correctly describes A?
If the eigenvalues of matrix A are all equal to 2, which of the following correctly describes A?
What property relates the eigenvalues of the adjugate matrix Adj A to those of matrix A?
What property relates the eigenvalues of the adjugate matrix Adj A to those of matrix A?
For a lower triangular matrix, how are the eigenvalues determined?
For a lower triangular matrix, how are the eigenvalues determined?
Given that the determinant of matrix A is 4, which of the following statements about eigenvalues is true?
Given that the determinant of matrix A is 4, which of the following statements about eigenvalues is true?
What condition must hold for a matrix A to be diagonalizable?
What condition must hold for a matrix A to be diagonalizable?
Given the eigenvalue λ = 5, which of the following represents an eigenvector for matrix A if x = t [1]?
Given the eigenvalue λ = 5, which of the following represents an eigenvector for matrix A if x = t [1]?
For which value of λ does the matrix A have an algebraic multiplicity of 2 and a geometric multiplicity of 2?
For which value of λ does the matrix A have an algebraic multiplicity of 2 and a geometric multiplicity of 2?
What does the equation D = P⁻¹AP signify in relation to a diagonalizable matrix?
What does the equation D = P⁻¹AP signify in relation to a diagonalizable matrix?
In the example provided, which eigenvalue has a corresponding eigenvector expressed as t(1, 1) for matrix A?
In the example provided, which eigenvalue has a corresponding eigenvector expressed as t(1, 1) for matrix A?
What is the characteristic polynomial for the matrix A in the last example?
What is the characteristic polynomial for the matrix A in the last example?
Under what condition is the matrix A considered not diagonalizable in the last example?
Under what condition is the matrix A considered not diagonalizable in the last example?
What is the relationship between x₁, x₂, and x₃ in the example with λ = 4?
What is the relationship between x₁, x₂, and x₃ in the example with λ = 4?
What does the Cayley Hamilton Theorem state regarding matrix A?
What does the Cayley Hamilton Theorem state regarding matrix A?
Which term correctly represents the eigenvalue λ associated with matrix A in the example if λ = 9 and is repeated thrice?
Which term correctly represents the eigenvalue λ associated with matrix A in the example if λ = 9 and is repeated thrice?
If the equation $ ext{sin } A = pA + qI$ is valid, what can be inferred about the relationship of $p$ and $q$ when $ ext{sin } π = 0$?
If the equation $ ext{sin } A = pA + qI$ is valid, what can be inferred about the relationship of $p$ and $q$ when $ ext{sin } π = 0$?
What is the result of simplifying $e^{1t} + e^{-1t} - e^{1t} - e^{-1t} + 1 = 0$?
What is the result of simplifying $e^{1t} + e^{-1t} - e^{1t} - e^{-1t} + 1 = 0$?
For the function $ ext{sin } λ = pλ + q$, what is the value of $p$ when $λ = 3π/2$?
For the function $ ext{sin } λ = pλ + q$, what is the value of $p$ when $λ = 3π/2$?
What roots does the cubic polynomial $λ^3 - 6λ^2 + 9λ - 4 = 0$ yield?
What roots does the cubic polynomial $λ^3 - 6λ^2 + 9λ - 4 = 0$ yield?
Considering the equation $λ^2 + 1 = 0$, what values does $λ$ take?
Considering the equation $λ^2 + 1 = 0$, what values does $λ$ take?
What is the significance of the formula $e^{tA} = pA + qI$ in terms of matrix exponentials?
What is the significance of the formula $e^{tA} = pA + qI$ in terms of matrix exponentials?
If $A = egin{bmatrix} 3 & 4 & -1 \ 5 & -2 & 1 \ 2 & 1 & 3 \ ext{ and } A^{-1} = rac{1}{5} egin{bmatrix} 2 & 4 & -1 \ 1 & 3 & -1 \ -1 & -1 & 2 \ ext{, what is the relationship of the matrices } A ext{ and } A^{-1}?$
If $A = egin{bmatrix} 3 & 4 & -1 \ 5 & -2 & 1 \ 2 & 1 & 3 \ ext{ and } A^{-1} = rac{1}{5} egin{bmatrix} 2 & 4 & -1 \ 1 & 3 & -1 \ -1 & -1 & 2 \ ext{, what is the relationship of the matrices } A ext{ and } A^{-1}?$
When solving for $q$ from the equation $3π/2 + q = -1$, what is the correct value?
When solving for $q$ from the equation $3π/2 + q = -1$, what is the correct value?
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Study Notes
Eigenvalues and Eigenvectors
- The characteristic polynomial of a square matrix A is defined as
f(λ) = det(A - λI), whereλis an eigenvalue andIis the identity matrix. - The characteristic equation is found by setting the characteristic polynomial equal to zero:
det(A - λI) = 0. - The roots of the characteristic equation are the eigenvalues of the matrix.
- To calculate the eigenvalues of a matrix, find the determinant of
(A - λI), set the determinant equal to zero, and solve the resulting equation forλ. - The eigenvalues of a matrix can be used to determine the stability of a system.
Properties of Eigenvalues
- The sum of the eigenvalues of a matrix is equal to the trace of the matrix.
- The product of the eigenvalues of a matrix is equal to the determinant of the matrix.
- For a 2x2 matrix, the characteristic equation is
λ² - (trace A)λ + det A = 0. - For a 3x3 matrix, the characteristic equation is
λ³ - (trace A)λ² + (c₁₁ + c₂₂ + c₃₃)λ - det A = 0, wherecᵢⱼare the elements of the matrix. - If A is a diagonal matrix, an upper triangular matrix, or a lower triangular matrix, then the diagonal elements are the eigenvalues of A.
- If
λis an eigenvalue of A, theng(λ)is an eigenvalue ofg(A), whereg(x)is any polynomial. - The eigenvalues of
Adj Aaredet(A)/λ.
Eigenvectors
- An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, is simply scaled by a factor equal to the corresponding eigenvalue.
- To find the eigenvectors of a matrix, solve the equation
(A - λI)x = 0, whereλis an eigenvalue andxis the eigenvector.
Diagonalizable Matrices
- A matrix is said to be diagonalizable if it can be transformed into a diagonal matrix by a similarity transformation.
- A matrix is diagonalizable if and only if the algebraic multiplicity of each eigenvalue is equal to its geometric multiplicity.
- If a matrix A is diagonalizable, there exists a diagonal matrix D and a diagonalizing matrix P such that
D = P⁻¹AP. - A matrix with distinct eigenvalues is always diagonalizable.
Examples
-
Example 1:
A = [ 1 4 -2 ][ -2 4 -2 ][ -2 4 2 ] -
For this matrix, the characteristic equation is
λ³ - 4λ² + 6λ - 6 = 0. -
The algebraic multiplicity of the eigenvalue 9 is 2, and the geometric multiplicity is also 2, meaning the matrix is diagonalizable.
-
The diagonal matrix D is
[9 0 0].[ 0 18 0][ 0 0 18] -
The diagonalizing matrix P is
[ -2 -1 1 ].[ 0 1 2 ][ 1 2 0 ] -
Example 2:
A = [ 1 3 7 ][ 4 2 3 ][ 1 1 2 ] -
The characteristic equation is
λ³ - 4λ² - 20λ - 35 = 0. -
The algebraic multiplicity of each eigenvalue is 1, but the geometric multiplicity is only 1.
-
This matrix is not diagonalizable.
Cayley-Hamilton Theorem
- The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation.
- This theorem can be used to find the inverse of a matrix, calculate powers of a matrix, and solve systems of differential equations.
Application: Finding the Sine of a Matrix
- If A is an upper triangular matrix, then
sin A = pA + qI, wherepandqare constants. - To find
pandq, we can use the following:sin λ = pλ + q- Substitute specific values of
λto solve forpandq.
Application: Exponential of a Matrix
- The exponential of a matrix is defined as
eᵗᴬ = pA + qI, wherepandqare constants. - To find
pandq, we can use the following:e¹ᵗ = pλ + q- Use properties of exponential functions and substitute appropriate values of
λandt.
Application: Inverse of a Matrix
- The inverse of a matrix can be found using the formula
A⁻¹ = Adj(A) / det(A), whereAdj(A)is the adjugate of A.
Key Examples
- The text includes examples of calculating eigenvalues and eigenvectors, and determining diagonalizability.
- These examples demonstrate the core concepts and provide practical applications for understanding the theory.
- The text also includes examples of finding the inverse of a matrix and calculating the sine and exponential of a matrix.
- These examples highlight important applications of eigenvalues and eigenvectors in linear algebra.
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