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Chapter 7: Problem 8

Let \(\|\cdot\|_{M}\) denote a matrix norm on \(\mathbb{R}^{n \times n},\|\cdot\|_{V}\) denote a vector norm on \(\mathbb{R}^{n},\) and \(I\) be the \(n \times n\) identity matrix. Show that (a) If \(\|\cdot\|_{M}\) and \(\|\cdot\|_{V}\) are compatible, then \(\|I\|_{M} \geq\) 1 (b) If \(\|\cdot\|_{M}\) is subordinate to \(\|\cdot\|_{V},\) then \(\|I\|_{M}=1\)

Short Answer

Expert verified
In summary, for compatible norms \(\|I\|_{M} \geq 1\) and for subordinate norms \(\|I\|_{M} = 1\).

Step by step solution

01

Apply the Compatibility Condition on the Identity Matrix

Since the matrix and vector norms are compatible, use the definition of compatibility and apply it to the identity matrix, I. Let \(A = I\) and \(x\neq0\). That is: $$\|Ix\|_{V} \leq \|I\|_{M} \|x\|_{V}.$$
02

Simplify the Left-Hand Side of the Inequality

Since \(I\) is the identity matrix, we know that \(Ix = x\). Thus, the inequality becomes: $$\|x\|_{V} \leq \|I\|_{M} \|x\|_{V}.$$
03

Analyze the Inequality

Since \(x\neq0\), we can now divide both sides of the inequality by \(\|x\|_{V}\): $$\frac{\|x\|_{V}}{\|x\|_{V}} \leq \|I\|_{M}.$$ This reduces the inequality to: $$1 \leq \|I\|_{M}.$$ Hence, for compatible norms, \(\|I\|_{M} \geq 1\). (b) Subordinate Norms and \(\|I\|_{M} = 1\).
04

Apply the Subordinate Condition on the Identity Matrix

Since the matrix norm is subordinate to the vector norm, use the definition of subordinate and apply it to the identity matrix, I: $$\|I\|_{M} = \sup_{x\neq0} \frac{\|Ix\|_{V}}{\|x\|_{V}}.$$
05

Simplify the Right-Hand Side of the Equation

Again, because I is the identity matrix, we know that \(Ix = x\). Thus the equation becomes: $$\|I\|_{M} = \sup_{x\neq0} \frac{\|x\|_{V}}{\|x\|_{V}}.$$
06

Evaluate the Supremum

Since \(x\neq0\), the term inside the supremum reduces to: $$\frac{\|x\|_{V}}{\|x\|_{V}} = 1.$$ Since there is no greater value than 1, we can conclude that: $$\|I\|_{M} = 1.$$ Hence, for subordinate norms, \(\|I\|_{M} = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Norm
A vector norm is a mathematical tool that measures the size, or length, of a vector in a vector space. Imagine you have an arrow on a graph; the vector norm tells you how long this arrow is.

Here are some important points about vector norms:
  • They are denoted by \| \cdot \|_V, where \(V\) stands for the vector norm.
  • A vector norm is a function that assigns a non-negative number to each vector.
  • It satisfies three properties: positivity, scalability, and the triangle inequality.
Intuitively, a vector norm helps you determine how far away a point (represented by a vector) is from the origin in a multi-dimensional space. Common vector norms include the Euclidean Norm, which is like measuring the straight-line distance between two points, and the Manhattan Norm, which measures the path distance along grid lines.
Identity Matrix
The identity matrix, often denoted as \(I\), serves as a neutral building block in matrix arithmetic, similar to the number one in multiplication.

Here's what makes the identity matrix special:
  • It is a square matrix, meaning it has the same number of rows and columns, typically \(n \times n\).
  • The main diagonal of \(I\) is filled with ones, while all other elements are zeros.
  • When any matrix \(A\) is multiplied by \(I\), the result is the same matrix \(A\). In other words, \(AI = IA = A\).
This property makes the identity matrix pivotal when exploring concepts like compatibility and subordinacy in matrix norms. Think of it as the matrix equivalent of the number one鈥攁lmost invisible when in operations, yet crucial for defining properties and understanding behaviors of other matrices.
Compatibility Condition
The compatibility condition connects vector norms and matrix norms by setting a rule about how they should interact when multiplying vectors by matrices. If two norms are compatible, you can control how matrix transformations affect vector lengths.

Key features of this condition include:
  • It ensures that transforming a vector by a matrix doesn't increase its norm disproportionately.
  • The relation is typically expressed as \[ \| Ax \|_V \leq \| A \|_M \| x \|_V, \]\ where \( A \) is a matrix, \( x \) is a vector, \( \| \cdot \|_V \) is the vector norm, and \( \| \cdot \|_M \) is the matrix norm.
This condition maintains a balance in computations, ensuring that the matrix norm acts as a bound on how much the matrix operation can "stretch" the vector. For the identity matrix, this results in the statement that \( \|I\|_M \geq 1 \). It implies that unit vectors remain unit or get expanded if the norms are governed by this compatibility.
Subordinate Norm
A matrix norm is termed as subordinate to a vector norm if it behaves in a way that naturally aligns with the vector norm when transforming vectors. This often helps streamline calculations by imposing a sense of consistency across different norm types.

Some key aspects of subordinate norms include:
  • They are defined using the supremacy of vector transformations: \[ \|A\|_M = \sup_{x eq 0} \frac{\|Ax\|_V}{\|x\|_V}.\]
  • The identity matrix always has a subordinate norm equal to 1, denoting no unnatural change in length for non-zero vectors.
  • This system creates a natural limit on how much the matrix can "act" on any given vector, echoing the vector鈥檚 properties.
Subordinate norms work as a reflection of the vector norm within matrix interactions, emphasizing the fact that these norms can be trusted to preserve or appropriately scale vector properties throughout transformations. When eigenvalues are less than or equal to one, this relationship further strengthens the condition \( \|I\|_M = 1 \).

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