Homework Answers
I have done all parts for you. Kindly go through. I have
explained in detail the steps and in b part I have given the reason
of how we reached in to the conclusion of the eigen vectors of 1
and -1. The third part is easy since we have a theorem which says
that a square matrix A is diagonalisable iff the direct sum of
eigen spaces of A is the whole vector space. So here we proved that
is a direct sum of the eigen spaces of A. Hence A is
diagonalisable.
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b)...
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b) If AT = A and if vectors u and v satisfy Au = 3u and Av = 40, then u: v=0. (c) An n x n symmetric matrix has n distinct real eigenvalues. (d) For a nonzero v in R", the matrix vv7 is a rank-1 matrix.
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,
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
linear algebra (1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A,...
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent....
estion 3 Let A be an n x n symmetric matrix. Then, which of the following...
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any XER, we can write A= XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mn.n(R). That is to say, V is a subspace, and V #Mnn(R) (there is some Me M.,n(R) such that M&V). Show that there exists an invertible matrix M e Mn.n(R) such...
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...
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0....
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0. (b) Show that there is no 5 x 5 matrix B such that B2 + 15 = 0. (c) Let C be any n xn matrix such that C2 + In 0. Let l be any eigenvalue of C. Show that 12 Conclude that C has no real eigenvalues. [1] [3] =-1. [3]
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b) If AT = A and if vectors u and v satisfy Au = 3u and Av = 40, then u: v=0. (c) An n x n symmetric matrix has n distinct real eigenvalues. (d) For a nonzero v in R", the matrix vv7 is a rank-1 matrix.
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,
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent....
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any XER, we can write A= XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mn.n(R). That is to say, V is a subspace, and V #Mnn(R) (there is some Me M.,n(R) such that M&V). Show that there exists an invertible matrix M e Mn.n(R) such...
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...
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0. (b) Show that there is no 5 x 5 matrix B such that B2 + 15 = 0. (c) Let C be any n xn matrix such that C2 + In 0. Let l be any eigenvalue of C. Show that 12 Conclude that C has no real eigenvalues. [1] [3] =-1. [3]
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