Question

Let \(A\) be any \(m \times n\) matrix and write \(K=\left\\{\mathbf{x} \mid A^{T} A \mathbf{x}=\mathbf{0}\right\\} .\) Let \(\mathbf{b}\) be an \(m\) -column. Show that if \(\mathbf{z}\) is an \(n\) -column such that \(\|\mathbf{b}-A \mathbf{z}\|\) is minimal, then all such vectors have the form \(\mathbf{z}+\mathbf{x}\) for some \(\mathbf{x} \in K\). [Hint: \(\|\mathbf{b}-A \mathbf{y}\|\) is minimal if and only if \(\left.A^{T} A \mathbf{y}=A^{T} \mathbf{b} .\right]\)

Short answer

Vectors minimizing \( \|\mathbf{b} - A\mathbf{z}\| \) are \( \mathbf{z} + \mathbf{x} \) with \( \mathbf{x} \in K \).

Step-by-step solution

Understanding the Set K

First, observe that the set \( K \) is defined as \( K = \{ \mathbf{x} \mid A^T A \mathbf{x} = \mathbf{0} \} \). This is the null space of the matrix \( A^T A \). It consists of all vectors that become zero when multiplied by \( A^T A \).

Identifying Conditions for Minimization

According to the hint, a vector \( \mathbf{y} \) makes \( \| \mathbf{b} - A \mathbf{y} \| \) minimal if and only if \( A^T A \mathbf{y} = A^T \mathbf{b} \). This condition defines the vectors \( \mathbf{y} \) in a way that the residual \( \mathbf{b} - A \mathbf{y} \) is perpendicular to the column space of \( A \).

Expressing the Solution Vector

Since \( \mathbf{z} \) is a vector such that \( \| \mathbf{b} - A \mathbf{z} \| \) is minimal, it follows that \( A^T A \mathbf{z} = A^T \mathbf{b} \). If you have another vector \( \mathbf{z}' \) that also satisfies this minimal condition, then \( A^T A \mathbf{z}' = A^T \mathbf{b} \).

Relating the Two Vectors

Since both \( \mathbf{z} \) and \( \mathbf{z}' \) satisfy the condition \( A^T A \mathbf{y} = A^T \mathbf{b} \), it implies that \( A^T A (\mathbf{z} - \mathbf{z}') = \mathbf{0} \). Therefore, \( \mathbf{z} - \mathbf{z}' \in K \).

Form of Solution Vectors

Thus, any vector \( \mathbf{z}' \) that makes the norm \( \| \mathbf{b} - A \mathbf{z}' \| \) minimal can be written as \( \mathbf{z}' = \mathbf{z} + \mathbf{x} \) for some \( \mathbf{x} \in K \). Hence, all such vectors have this form.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing a third matrix by combining elements from the original matrices in a specific way.

• When we multiply an \( m \times n \) matrix \( A \) by an \( n \times p \) matrix \( B \), the result is an \( m \times p \) matrix.
• The elements of the resulting matrix are computed by taking the dot product of rows from \( A \) with columns from \( B \).
• Matrix multiplication is not commutative, which means \( AB \) does not necessarily equal \( BA \).

In the context of the problem, we are particularly interested in multiplying a matrix by its transpose, which is crucial for understanding the null space where the minimization problem occurs. For instance, \( A^T A \) is key to deriving conditions for minimization in least squares problems.

Orthogonality

Orthogonality is a concept that describes a perpendicular relationship between two vectors. This means that the dot product of two orthogonal vectors is zero.

• Orthogonal vectors are important in determining the solution to problems as they often represent independent directions in space.
• In our context, once you know that \( \| \mathbf{b} - A \mathbf{y} \| \) is minimized, it implies that the vector \( \mathbf{b} - A \mathbf{y} \) is orthogonal to the column space of \( A \).
• Orthogonality is leveraged to find unique solutions in problems and ensure the minimal distance condition is met.

Understanding orthogonality helps recognize why the difference between vectors that meet the minimization condition must lie in the null space, thus explaining why a solution can be expressed as \( \mathbf{z} + \mathbf{x} \).

Minimization

Minimization in this exercise involves finding a vector \( \mathbf{z} \) where the distance between \( \mathbf{b} \) and \( A\mathbf{z} \) is as small as possible. This distance is defined by the norm \( \| \mathbf{b} - A\mathbf{z} \| \).

• A key condition for this minimization is that \( A^T A \mathbf{z} = A^T \mathbf{b} \).
• This equation ensures that the vector difference \( \mathbf{b} - A\mathbf{z} \) is minimized or as small as possible.
• Every solution to this optimization problem can be expressed as a sum of a particular solution and a vector from the null space, showing flexibility in solutions.

The minimization concept ties back to finding solutions efficiently in systems where direct computation might be challenging. Recognizing how minimization is connected to orthogonality and the null space is essential for solving related linear algebra problems.