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1. Find a 2x2 matrix A if for the vector v= [R], Av = [4 +38]...
1. Find a 2x2 matrix A if for the vector v= [R], Av = [4 +38] 2. For this problem, use matrices A = La ), B=1 _Jandc=lo 9]. Suppose that the matrices A and B commute (so AB=BA) and the matrices A and C commute. Find the entries for the matrix A. 3. Find a number a so that the vectors v = [3 2 a) and w = [2a -1 3] are orthogonal (perpendicular). 4. For the vector...
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2....
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2. Hint: What is the matrix of the projection onto the coordinate axis x1? Problem 5. Problem 5.4 on page 23. The following method is suggested: (1) Find an angle o such that the line x1 = –2x2 is obtained by rotating the x-axis by 0. (2) Convince yourself with geometry that to project a vector v onto the line x1 = –2x2 is the...
Find the steady state probability vector for the matrix. An eigenvector v of an n ×...
Find the steady state probability vector for the matrix. An
eigenvector v of an n × n matrix
A is a steady state probability vector when
Av = v and the
components of v sum to 1.
Find the steady state probability vector for the matrix. An eigenvector v of an n x n matrix A is a steady state probability vector when Av = v and the components of v sum to 1. 0.9 0.4 A = 0.1 0.6
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic fi...
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
4. Let A be an n x n matrix. Define the trace of A by the...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
(1 point) A square matrix A is idempotent if A2 = A. Let V be the...
(1 point) A square matrix A is idempotent if A2 = A. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in...
a) Find the eigenvalues and the eigenvectors of the 2x2 matrix: [4 2] [3 -1] b)...
a) Find the eigenvalues and the eigenvectors of the 2x2 matrix: [4 2] [3 -1] b) Solve the initial value problem: dx/dt = 4x + 2y dy/dt = 3x - y with x(0) = 0, y(0) = 7
Let D be a 2x2 linear transformation matrix that transforms the vector ? = [ 1...
Let D be a 2x2 linear transformation matrix that transforms the vector ? = [ 1 4 ] into the vector ?? = [ 3 6 ] and transforms the vector ? = [ 2 5 ] into the vector ?? = [ 0 9 ]. Analyze the linear transformation matrix D by doing the following: Let S be a square with side length 2, located in the xy-plane. The matrix D transforms the vertices of S into the vertices...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetr...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What...
linear algebra Find the steady state probability vector for the matrix. An eigenvector v of an...
linear algebra
Find the steady state probability vector for the matrix. An eigenvector v of an nxn matrix A is a steady state probability vector when Av = v and the components of v sum to 1. 0.8 0.3 A = 0.2 0.7
1. Find a 2x2 matrix A if for the vector v= [R], Av = [4 +38] 2. For this problem, use matrices A = La ), B=1 _Jandc=lo 9]. Suppose that the matrices A and B commute (so AB=BA) and the matrices A and C commute. Find the entries for the matrix A. 3. Find a number a so that the vectors v = [3 2 a) and w = [2a -1 3] are orthogonal (perpendicular). 4. For the vector...
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2. Hint: What is the matrix of the projection onto the coordinate axis x1? Problem 5. Problem 5.4 on page 23. The following method is suggested: (1) Find an angle o such that the line x1 = –2x2 is obtained by rotating the x-axis by 0. (2) Convince yourself with geometry that to project a vector v onto the line x1 = –2x2 is the...
Find the steady state probability vector for the matrix. An
eigenvector v of an n × n matrix
A is a steady state probability vector when
Av = v and the
components of v sum to 1.
Find the steady state probability vector for the matrix. An eigenvector v of an n x n matrix A is a steady state probability vector when Av = v and the components of v sum to 1. 0.9 0.4 A = 0.1 0.6
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
(1 point) A square matrix A is idempotent if A2 = A. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What...
linear algebra
Find the steady state probability vector for the matrix. An eigenvector v of an nxn matrix A is a steady state probability vector when Av = v and the components of v sum to 1. 0.8 0.3 A = 0.2 0.7
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