A = 5 0 0 0 -6 -1 -1 0060 0 0 -11 -5 - 11
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Step by step
Solved in 7 steps with 7 images
- Show that A=[0110] has no real eigenvalues.Write the system of differential equations given in the 1st photo in the normal form defined in the 2nd photo and solve the resulting system by means of the eigenvalues and eigen-vectors of the square matrix A.Suppose that all the eigenvalues of the matrix A have negative real part. Then everysolution of the differential equation x`= Ax satisfies,|x(t)| ≤ |x(s)|, if t > s.
- For which value of k does the matrix have one real eigenvalue of multiplicity 2A is a 2 X 2 matrix with eigenvectors v1 and v2 corresponding to eigenvalues λ1 = 1/2and λ2 = 2, respectively, and x Find A kx. What happens as k becomes large (i.e., k--> ∞)you are given the eigenvalues and eigenvectors for a 2 × 2 matrixA. Write down the general solution of dx/dt = Ax
- solve the following system of linear non-homogeneous differential equations with eigenvalues/eigenvectors and variation of parametersA is a 3 X 3matrix with eigenvectors v1 ,v2 and v3 corresponding to eigenvalues λ1 = -1/3,λ2=1/3 and λ3 = 1, respectively, and x .Find A kx. What happens as k becomes large (i.e., k--->∞ )The 2×2 real symmetric matrix A has two distinct eigenvalues λ1 and λ2.If v1 =(1,2) is an eigenvector of A corresponding to the eigen value λ1,determine an eigenvector corresponding to λ2.
- What is the relationship between the characteristic polynomial and the eigenvalues of A?Here we have a system of differential equations. (dx/dt) = 11x + 8y, (dy/dt) = 8x - y. The coefficient matrix, [11 8][8 -1] has eigenvectors: [1] and [2] [-2] [1] With corresponding eigenvalues of -5 and 15. what is the general solution of this system?When S and T are symmetric positive definite, ST might not even be symmetric. But its eigenvalues are still positive. Start from STx = AX and take dot products with Tx. Then prove A > 0.
