Product of pseudo-inverses :Suppose A and D are right-invertible matrices and the product AD exists. We...
Product of pseudo-inverses :Suppose A and D are right-invertible matrices and the product AD exists. We have seen that if B is a right inverse of A and E is a right inverse of D, then EB is a right inverse of AD. Now suppose B is the pseudo-inverse of A and E is the pseudo-inverse of D. Is EB the pseudo-inverse of AD? Prove that this is always true or give an example for which it is false.
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Product of pseudo-inverses :Suppose A and D are right-invertible
matrices and the product AD exists. We...
Product of pseudo-inverses Suppose A and D are right-invertible matrices and the prod- uct AD exists. We have seen that if B is a right inverse of A and E is a right inverse of D, then EB is a right i...
Product of pseudo-inverses Suppose A and D are right-invertible
matrices and the prod-
uct AD exists. We have seen that if B is a right inverse of A
and E is a right inverse of D, then EB
is a right inverse of AD. Now suppose B is the pseudo-inverse of
A and E is the pseudo-inverse of
D. Is EB the pseudo-inverse of AD? Prove that this is always
true or give an example for which it
is false....
11.23 Product of pseudo-inverses. Suppose A and D are right-invertible matrices and the prod- uct AD...
11.23 Product of pseudo-inverses. Suppose A and D are right-invertible matrices and the prod- uct AD exists. We have seen that if B is a right inverse of A and E is a right inverse of D, then EB is a right inverse of AD. Now suppose B is the pseudo-inverse of A and E is the pseudo-inverse of D. Is EB the pseudo-inverse of AD? Prove that this is always true or give an example for which it is...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is inv...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
3. (10 points) Simultaneous left inverse The two matrices 3 2] and both left-invertible, and have multiple left inverses. Do they have a common left inverse? Explain how to find a 2 × 4 matrix C that...
3. (10 points) Simultaneous left inverse The two matrices 3 2] and both left-invertible, and have multiple left inverses. Do they have a common left inverse? Explain how to find a 2 × 4 matrix C that satisfies CA-CB-1, or determine that no such matrix exists. (You can use numerical computing to find C.) Hint. Set up a set of linear equations for the entries of C. Remark. There is nothing special about the particular entries of the two matrices...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A € Mmxm(R) is the matrix A+ E Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+ AA+ = A+ (c) (AA+)? = AA+ (d) (A+A)T = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-I. The following is true: Any...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A E Mmxm(R) is the matrix At E Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+AA+ = A+ (c) (AA+)T = AA+ (d) (A+A)T = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-1. The following is true: Any matrix...
Verify the following properties, using any distinct, invertible A, B, 4×4 upper triangular matrices of your...
Verify the following properties, using any distinct, invertible
A, B, 4×4 upper triangular matrices of your choice:
3. (0.5 marks each) Verify the following properties, using any distinct, invertible A, B, 4 x 4 upper triangular matrices of your choice: (a) The inverse of an upper triangular matrix is upper triangular; (b) (AB)- B-1A-1 (e) trace(AB) trace(BA); (d) det(AB) det (BA) example of matrices A, B such that det(AB) det(BA) (BONUS 1 mark) Give an
3. (0.5 marks each) Verify...
We say that A and B are similar matrices if A = SBS-1 for some invertible...
We say that A and B are similar matrices if A = SBS-1 for some invertible matrix S. Are the following true or false. Given a mathematical reason (proof). (a) If A and B are similar, then A and B have the same eigenvalues. Answer: (b) If A and B are similar, then A and B have the same eigenvectors. Answer: c) If A and B are similar, then A - 51 and B – 51 are similar. Answer: (d)...
Decide whether each statement is true or false and explain your reasoning. Give a counter-example...
Decide whether each statement is true or false and explain your reasoning. Give a counter-example for false statements. The matrices A and B are n x n. a. The equation Ax b must have at least one solution for all b e R". b. IfAx-0 has only the trivial solution, then A is row equivalent to the n x p, identity matrix. c. If A is invertible, then the columns of A-1 are linearly independent. d. If A is invertible,...
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, ar...
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, are the columns of AB 1. Suppose A is an nxnmatrix such that A -SDS where D diag(di,d2,... dn) is a diagonal matrix, and S is an invertible matrix. Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D Before proving this, work through the following...
Product of pseudo-inverses Suppose A and D are right-invertible
matrices and the prod-
uct AD exists. We have seen that if B is a right inverse of A
and E is a right inverse of D, then EB
is a right inverse of AD. Now suppose B is the pseudo-inverse of
A and E is the pseudo-inverse of
D. Is EB the pseudo-inverse of AD? Prove that this is always
true or give an example for which it
is false....
11.23 Product of pseudo-inverses. Suppose A and D are right-invertible matrices and the prod- uct AD exists. We have seen that if B is a right inverse of A and E is a right inverse of D, then EB is a right inverse of AD. Now suppose B is the pseudo-inverse of A and E is the pseudo-inverse of D. Is EB the pseudo-inverse of AD? Prove that this is always true or give an example for which it is...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
3. (10 points) Simultaneous left inverse The two matrices 3 2] and both left-invertible, and have multiple left inverses. Do they have a common left inverse? Explain how to find a 2 × 4 matrix C that satisfies CA-CB-1, or determine that no such matrix exists. (You can use numerical computing to find C.) Hint. Set up a set of linear equations for the entries of C. Remark. There is nothing special about the particular entries of the two matrices...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A € Mmxm(R) is the matrix A+ E Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+ AA+ = A+ (c) (AA+)? = AA+ (d) (A+A)T = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-I. The following is true: Any...
9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A E Mmxm(R) is the matrix At E Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+AA+ = A+ (c) (AA+)T = AA+ (d) (A+A)T = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-1. The following is true: Any matrix...
Verify the following properties, using any distinct, invertible
A, B, 4×4 upper triangular matrices of your choice:
3. (0.5 marks each) Verify the following properties, using any distinct, invertible A, B, 4 x 4 upper triangular matrices of your choice: (a) The inverse of an upper triangular matrix is upper triangular; (b) (AB)- B-1A-1 (e) trace(AB) trace(BA); (d) det(AB) det (BA) example of matrices A, B such that det(AB) det(BA) (BONUS 1 mark) Give an
3. (0.5 marks each) Verify...
We say that A and B are similar matrices if A = SBS-1 for some invertible matrix S. Are the following true or false. Given a mathematical reason (proof). (a) If A and B are similar, then A and B have the same eigenvalues. Answer: (b) If A and B are similar, then A and B have the same eigenvectors. Answer: c) If A and B are similar, then A - 51 and B – 51 are similar. Answer: (d)...
Decide whether each statement is true or false and explain your reasoning. Give a counter-example for false statements. The matrices A and B are n x n. a. The equation Ax b must have at least one solution for all b e R". b. IfAx-0 has only the trivial solution, then A is row equivalent to the n x p, identity matrix. c. If A is invertible, then the columns of A-1 are linearly independent. d. If A is invertible,...
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, are the columns of AB 1. Suppose A is an nxnmatrix such that A -SDS where D diag(di,d2,... dn) is a diagonal matrix, and S is an invertible matrix. Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D Before proving this, work through the following...
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