Matrices
by definition, an nxn matrix A is nilpotent. if there is an integer k>0 such that A^k = 0, the zero matrix. Prove the following: If A is nilpotent, then I+A is nonsingular. (Here I is the indentity matrix of size nxn.)
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Matrix Problem
by definition, an nxn matrix A is unipotent. If I-A is nilpotent. Prove the following: If A is unipotent, then A is nonsingular.
Matrices
by definition, an nxn matrix T = (tᵢⱼ) is upper triangular if tᵢⱼ = 0 whenever i > j. Prove the following: If T is upper triangular and nilpotent, then tᵢᵢ = 0 for all i.
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent ...
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove...
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove the following statements (a) If A is nilpotent, then A 0. (Hint: First show that if A is nilpotent, then the Jordan form of A is also nilpotent.) (b) If A is nilpotent, then tr(A) 0 (e) A is nilpotent if and only if the characteristic polynomial of A is (-1)"" (d) If A is...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent matrix and use that along with the power series definition of the matrix exponential to find eAt 2!
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F)...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
2 0 0 2. Let A be the diagonal matrix 0 4 0First read Exercise 2...
2 0 0 2. Let A be the diagonal matrix 0 4 0First read Exercise 2 of Section 1.5, before continung (a) What would it mean to say that A is nonsingular? (b) Prove that A nonsingular. Give a full explanation using your definition in part Let A be a 4 × 4 matrix with its third row consisting of zeros. (a) What would it mean to say that A is nonsingular? (b) Prove that A is singular. (Hint: Exercise...
26) Prove that if A is a nonsing AB = AC, then B = C. Your...
26) Prove that if A is a nonsing AB = AC, then B = C. Your pro a nonsingular nxn matrix, and B and C are nxk matrices such that c. Your proof must be complete. (10 points) Proof:
a) suppose that the nxn matrix A has its n eigenvalues arranged in decreasing order of absolute size, so that >>....
a) suppose that the nxn matrix A has its n eigenvalues arranged
in decreasing order of absolute size, so that >>....
each eigenvalue has its corresponding eigenvector, x1,x2,...,xn.
suppose we make some initial guess y(0) for an eigenvector.
suppose, too, that y(0) can be written in terms of the actual
eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2
+...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants.
by considering the "power method" type iteration y(k+1)=Ay(k) argue
that (see attached image)
b) from an nxn...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove the following statements (a) If A is nilpotent, then A 0. (Hint: First show that if A is nilpotent, then the Jordan form of A is also nilpotent.) (b) If A is nilpotent, then tr(A) 0 (e) A is nilpotent if and only if the characteristic polynomial of A is (-1)"" (d) If A is...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent matrix and use that along with the power series definition of the matrix exponential to find eAt 2!
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
2 0 0 2. Let A be the diagonal matrix 0 4 0First read Exercise 2 of Section 1.5, before continung (a) What would it mean to say that A is nonsingular? (b) Prove that A nonsingular. Give a full explanation using your definition in part Let A be a 4 × 4 matrix with its third row consisting of zeros. (a) What would it mean to say that A is nonsingular? (b) Prove that A is singular. (Hint: Exercise...
26) Prove that if A is a nonsing AB = AC, then B = C. Your pro a nonsingular nxn matrix, and B and C are nxk matrices such that c. Your proof must be complete. (10 points) Proof:
a) suppose that the nxn matrix A has its n eigenvalues arranged
in decreasing order of absolute size, so that >>....
each eigenvalue has its corresponding eigenvector, x1,x2,...,xn.
suppose we make some initial guess y(0) for an eigenvector.
suppose, too, that y(0) can be written in terms of the actual
eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2
+...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants.
by considering the "power method" type iteration y(k+1)=Ay(k) argue
that (see attached image)
b) from an nxn...
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