Let A∈ℝnxn, and suppose ?,?∈ℝ with ?≠0 such that ?+?i is an eigenvalue of ?. Suppose...
Let A∈ℝnxn, and suppose ?,?∈ℝ with ?≠0 such that ?+?i is an eigenvalue of ?. Suppose vectors ?,?∈ℝ? such that ?+?? is an eigenvector for ? associated with the eigenvalue ?+?i. Prove that vectors ? and ? are linearly independent.
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Let A∈ℝnxn, and suppose ?,?∈ℝ with ?≠0 such that ?+?i
is an eigenvalue of ?. Suppose...
5. A is a nonsingular matrix (that is A-exists) and suppose is an eigenvalue of A...
5. A is a nonsingular matrix (that is A-exists) and suppose is an eigenvalue of A with associated eigenvector K. 5.1 Prove that 1 70. (Hint: Suppose that i = 0.) 5.2 Show that is an eigenvalue of A-- with corresponding eigenvector K. 5.3 Show that 12 is an eigenvalue of A² with corresponding eigenvector K. (This statement is true even if A is singular.)
Let v 2 Rn be a unit vector. Define G = I ? vvT . (a)...
Let v 2 Rn be a unit vector. Define G = I ? vvT .
(a) Show G is symmetric and G2 = G.
(b) Prove v is an eigenvector, find the associated
eigenvalue.
(c) Prove that if < u; v >= 0 then u is also an eigenvector
of G.
(d) Prove that G is diagonalizable.
Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c...
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an...
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an eigenvector of Mk where k is any positive integer? If so, what would the associated eigenvalue be? Now suppose that the matrix A is nilpotent, i.e. A* integer k 2. Show that 0 is the only eigenvalue of A. [Hint: what is det (A)? This should help you decide that A has an eigenvalue of 0 in particular. Then you need to demonstrate that...
Let , and let be a polynomial. Show that if is an eigenvalue of , then...
Let
, and let
be a polynomial. Show that if is an
eigenvalue of , then is an
eigenvalue of .
Hint: this follows from the more precise statement that if
is a
non-zero eigenvector for for the eigenvalue
, then is also an
eigenvector for for the
eigenvalue . Prove
this.
TEL(V) PEPF) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a)...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity an, and the geometric multiplicity, g, of each eigenvalue. (c) For one of the eigenvalues you should have gi < az. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of...
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma....
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
Let V be a vector space, and ffl, f2, fn) c V be linear functionals on V. Suppose we can find a vector vi e V such that fl (v) 6-0 but £2(v)-6(v) = . . .-m(v) = 0. Similarly, suppose that for all 1 i...
Let V be a vector space, and ffl, f2, fn) c V be linear functionals on V. Suppose we can find a vector vi e V such that fl (v) 6-0 but £2(v)-6(v) = . . .-m(v) = 0. Similarly, suppose that for all 1 i < n we can find vi є V such that fi(vi) 6-0 and fj (vi)-0 for alljöi. Prove that {fL-fa) is were linearly independent in V ly independent in V * . Prove also...
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue...
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue ofA is [7-3 1]T (a) Determine the eigenvalue associated with this eigenvector (b) For b-[1 1 1]T, find the approximate solution to b in the system x-Ab due to this eigenvector and compare it the exact solution.
5. A is a nonsingular matrix (that is A-exists) and suppose is an eigenvalue of A with associated eigenvector K. 5.1 Prove that 1 70. (Hint: Suppose that i = 0.) 5.2 Show that is an eigenvalue of A-- with corresponding eigenvector K. 5.3 Show that 12 is an eigenvalue of A² with corresponding eigenvector K. (This statement is true even if A is singular.)
Let v 2 Rn be a unit vector. Define G = I ? vvT .
(a) Show G is symmetric and G2 = G.
(b) Prove v is an eigenvector, find the associated
eigenvalue.
(c) Prove that if < u; v >= 0 then u is also an eigenvector
of G.
(d) Prove that G is diagonalizable.
Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
Suppose A is an eigenvalue of the matrix M with associated eigenvector v. Is v an eigenvector of Mk where k is any positive integer? If so, what would the associated eigenvalue be? Now suppose that the matrix A is nilpotent, i.e. A* integer k 2. Show that 0 is the only eigenvalue of A. [Hint: what is det (A)? This should help you decide that A has an eigenvalue of 0 in particular. Then you need to demonstrate that...
Let
, and let
be a polynomial. Show that if is an
eigenvalue of , then is an
eigenvalue of .
Hint: this follows from the more precise statement that if
is a
non-zero eigenvector for for the eigenvalue
, then is also an
eigenvector for for the
eigenvalue . Prove
this.
TEL(V) PEPF) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity an, and the geometric multiplicity, g, of each eigenvalue. (c) For one of the eigenvalues you should have gi < az. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of...
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
Let V be a vector space, and ffl, f2, fn) c V be linear functionals on V. Suppose we can find a vector vi e V such that fl (v) 6-0 but £2(v)-6(v) = . . .-m(v) = 0. Similarly, suppose that for all 1 i < n we can find vi є V such that fi(vi) 6-0 and fj (vi)-0 for alljöi. Prove that {fL-fa) is were linearly independent in V ly independent in V * . Prove also...
Given the matrix A467 333 0 2 0 0 The eigenvector associated with the largest eigenvalue ofA is [7-3 1]T (a) Determine the eigenvalue associated with this eigenvector (b) For b-[1 1 1]T, find the approximate solution to b in the system x-Ab due to this eigenvector and compare it the exact solution.
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