Topic
Moore-Penrose Pseudoinverse
Courses
Courses whose questions include this topic tag.
Questions
This topic is tagged for 10 questions
-
Fall 2021 • Question 2
(Moore-Penrose pseudo-inverse)
Let \(A \in \mathbb {C}^{m \times n}\) have rank \(r\) and reduced SVD \[ A = \tilde {U}\tilde {\Sigma }\tilde {V}^* . \]
The Moore-Penrose pseudoinverse of \(A\) is defined as
\[ A^+ = \tilde {V}\tilde {\Sigma }^{-1}\tilde {U}^* . \]a. In the induced matrix \(\ell ^2\) norm \(\|\cdot \|\), is the operation \(A \mapsto A^+\) well-conditioned? That is, given an arbitrary but fixed \(A\) and a perturbation matrix \(B\), is \(\|(A + B)^+ - A^+\|/\|A^+\|\) controllable by \(\|B\|/\|A\|\)? Prove it, or give a counterexample.
b. If \(m \ge n\) and \(A\) has full rank \(n\) and \(b \in \mathbb {C}^m\) is a given vector, prove that \(x = A^+b\) is identical to the least-squares solution \(z\) of the over-determined linear system \(Az = b\).
-
Fall 2013 • Question 2
**Least Squares Problems** **a)** For a real full rank \(m\times n\) matrix \(A\) and vector \(\mathbf{b}\in\mathbb{R}^m\), explain how to solve the least-squares problem of finding \(\mathbf{x}\in\mathbb{R}^n\) that minimizes \(\|A\mathbf{x}-\mathbf{b}\|_2\) using i) the normal equations, and b) a QR factorization of the matrix \(A\). What are the advantages and disadvantages of each of these methods? **b)** For a full rank real \(m\times n\) matrix \(A\), show that \(X=A^\dagger\), the pseudoinverse of \(A\), minimizes \(\|AX-I\|_F\) over all \(n\times m\) matrices \(X\). What is the value of the minimum? (Hint: Relate the problem to a set of least-squares problems).
-
Fall 2010 • Question 1
- State and prove the Fredholm Alternative Theorem for a linear transformation \(A:\mathbb {R}^n\to \mathbb {R}^m\).
- Find and give a graphical interpretation of the pseudo-inverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2 \end {pmatrix}. \]
-
Spring 2010 • Question 2
**Least Squares Problems** **a)** For a full rank real \(m\times n\) matrix \(A\), show that \(X=A^\dagger\), the pseudoinverse of \(A\), minimizes \(\|AX-I\|_F\) over all \(n\times m\) matrices \(X\). What is the value of the minimum? (Hint: Relate the problem to a set of least-squares problems). **b)** For a real full rank \(m\times n\) matrix \(A\) and vector \(b\in\mathbb{R}^m\), explain how to solve the least-squares problem of finding \(x\in\mathbb{R}^n\) that minimizes \(\|Ax-b\|_2\) using **i)** the normal equations, and **b)** a QR factorization of the matrix \(A\). What are the advantages and disadvantages of each of these methods?
-
Fall 2009 • Question 2
**Least Squares Problems** **a)** For a full rank real \(m\times n\) matrix \(A\), show that \(X=A^\dagger\), the pseudoinverse of \(A\), minimizes \(\|AX-I\|_F\) over all \(n\times m\) matrices \(X\). What is the value of the minimum? (Hint: Relate the problem to a set of least-squares problems). **b)** For a real full rank \(m\times n\) matrix \(A\) and vector \(b\in\mathbb{R}^m\), explain how to solve the least-squares problem of finding \(x\in\mathbb{R}^n\) that minimizes \(\|Ax-b\|_2\) using **i)** the normal equations, and **b)** a QR factorization of the matrix \(A\). What are the advantages and disadvantages of each of these methods?
-
Spring 2009 • Question 1
- The least squares psuedo-inverse of \(A\) is a matrix \(B\) which satisfies what two properties?
- Find the pseudoinverse of the matrix \[ A= \begin {pmatrix} 1 & 2\\ 1 & 2 \end {pmatrix}. \]
-
Fall 2008 • Question 1
Find a singular value decomposition and pseudoinverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2\\ 0 & 0 \end {pmatrix}. \]
-
Fall 1996 • Question A.3
Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\) matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).
- State the conditions that define \(A'\).
- Show that \(A'\) always exists.
- Construct \(A'\) explicitly for the \(2\times 2\) matrix \[ A=\begin {pmatrix} 1 & 1\\ 0.5 & 0.5 \end {pmatrix}. \]
-
Fall 1995 • Question A2
Let \(A\) be a real \(m\times n\) matrix.
- Define the Least Square Pseudo-Inverse \(A'\).
- Show how Gaussian Elimination can be used to compute \(A'\).
-
Fall 1994 • Question A2
Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\) matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).
- State the conditions that define \(A'\).
- Show that \(A'\) always exists.
- Construct \(A'\) explicitly for the \(2\times 2\) matrix \[ A=\begin {pmatrix} 1 & 1\\ 0 & 0 \end {pmatrix}. \]