Question

Show that the following are equivalent for an \(n \times n\) matrix \(P\). a. \(P\) is orthogonal. b. \(\|P \mathbf{x}\|=\|\mathbf{x}\|\) for all columns \(\mathbf{x}\) in \(\mathbb{R}^{n}\). c. \(\|P \mathbf{x}-P \mathbf{y}\|=\|\mathbf{x}-\mathbf{y}\|\) for all columns \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{R}^{n}\) d. \((P \mathbf{x}) \cdot(P \mathbf{y})=\mathbf{x} \cdot \mathbf{y}\) for all columns \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{R}^{n}\). [Hints: For \((\mathrm{c}) \Rightarrow(\mathrm{d}),\) see Exercise \(5.3 .14(\mathrm{a}) .\) For (d) \(\Rightarrow\) (a), show that column \(i\) of \(P\) equals \(P \mathbf{e}_{i}\), where \(\mathbf{e}_{i}\) is column \(i\) of the identity matrix. \(]\)

Short answer

The properties (a)-(d) for matrix \( P \) are logically equivalent.

Step-by-step solution

Prove (a) implies (b)

An orthogonal matrix satisfies the property \( P^TP = I \). This means for any vector \( \mathbf{x} \), we have \( \| P \mathbf{x} \|^2 = (P\mathbf{x})^T (P\mathbf{x}) = \mathbf{x}^T P^T P \mathbf{x} = \mathbf{x}^T I \mathbf{x} = \| \mathbf{x} \|^2 \). Thus, \( \| P \mathbf{x} \| = \| \mathbf{x} \| \), proving (b).

Prove (b) implies (c)

Given \( \|P \mathbf{x}\| = \|\mathbf{x}\| \) for any vector \( \mathbf{x} \), consider two vectors \( \mathbf{x} \) and \( \mathbf{y} \). By substituting \( \mathbf{x'} = \mathbf{x} - \mathbf{y} \), obtain \( \| P(\mathbf{x} - \mathbf{y}) \| = \| \mathbf{x} - \mathbf{y} \| \), showing \( \| P\mathbf{x} - P\mathbf{y} \| = \| \mathbf{x} - \mathbf{y} \| \), hence proving (c).

Prove (c) implies (d)

Assume \( \| P \mathbf{x} - P \mathbf{y} \| = \| \mathbf{x} - \mathbf{y} \| \) is true for all vectors \( \mathbf{x}, \mathbf{y} \). Squaring both sides results in \( (P\mathbf{x} - P\mathbf{y}) \cdot (P\mathbf{x} - P\mathbf{y}) = (\mathbf{x} - \mathbf{y}) \cdot (\mathbf{x} - \mathbf{y}) \). This condition implies: \[(P\mathbf{x}) \cdot (P\mathbf{y}) = \mathbf{x} \cdot \mathbf{y} \] Thus confirming (d).

Prove (d) implies (a)

Using \((P \mathbf{x}) \cdot (P \mathbf{y}) = \mathbf{x} \cdot \mathbf{y}\) for all vectors \( \mathbf{x}, \mathbf{y} \), consider the standard basis vector \( \mathbf{e}_i \). Then \( P\mathbf{e}_i \) produces column \( i \) of matrix \( P \). Since this operation keeps the dot product invariant, each column vector \(( P\mathbf{e}_i )\) of \( (P) \) is orthonormal. This means \( P \) is orthogonal and satisfies \( P^T P = I \), proving (a).

Conclusion: Equivalence Proof Completion

Starting from (a), which directly led to (b), (b) leading to (c), (c) resulting in (d), and (d) proving back to (a), all the properties were shown logically compact into one another. Thus, the equivalence of all four properties of the matrix \( P \) has been demonstrated.

Key concepts

Equivalence of Conditions

When we talk about the equivalence of conditions for an orthogonal matrix, we're referring to the ability of different mathematical statements to express the same underlying truth about the matrix. In the context of our problem, we are given four different statements about an \(n \times n\) matrix \(P\) and need to show they are equivalent. This means that if any one statement is true, then all the others must also be true.

The first statement is that \(P\) is orthogonal, meaning its transpose is also its inverse, or mathematically expressed as \(P^T P = I\), where \(I\) is the identity matrix. The second statement involves vector norms, claiming the transformation by \(P\) does not change the length of any vector it multiplies. The third condition extends this concept from individual vectors to vector differences, maintaining the "distance" or norm of any two vectors after transformation remains constant.

The final condition focuses on the dot product, asserting that \(P\) preserves the dot product of any two vectors, maintaining their angle and therefore their orthogonality or direction relative to each other. Demonstrating the equivalence of these conditions involves showing a logical flow where each one implies the next in a cycle, ultimately proving they all boil down to the fact \(P\) is orthogonal.

Vector Norms

Vector norms are a measure of the length or magnitude of a vector. In our problem, understanding vector norms is essential because it helps to prove that orthogonal matrices do not change the length of vectors they transform. The notation \(\|\mathbf{x}\|\) represents the norm of a vector \(\mathbf{x}\).

The crucial property of orthogonal matrices is that they preserve vector norms. For any vector \(\mathbf{x}\) in \(\mathbb{R}^n\), if \(P\) is orthogonal, it means \(\|P\mathbf{x}\| = \|\mathbf{x}\|\). But why is this important? It signifies that multiplying a vector by an orthogonal matrix does not stretch or shrink it; instead, the vector remains the same length, even if its direction might change.

Understanding this property allows us to extend to vector differences as well. For instance, if \(\mathbf{x}\) and \(\mathbf{y}\) are vectors, then \(\|P\mathbf{x} - P\mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\|\). This tells us that the transformation by \(P\) does not alter the distance between any two vectors—another hallmark of orthogonal matrices.

Dot Product

The dot product is a fundamental operation in vector algebra, representing an algebraic operation that combines two sequences of numbers into a single number. For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), their dot product is denoted \( \mathbf{a} \cdot \mathbf{b} \) and is calculated as \(a_1b_1 + a_2b_2 + \dots + a_nb_n\).

When it comes to orthogonal matrices, the property of preserving the dot product is significant. If \(P\) is an orthogonal matrix, it implies the dot product between any two transformed vectors remains the same as the dot product between the original vectors: \((P \mathbf{x}) \cdot (P \mathbf{y}) = \mathbf{x} \cdot \mathbf{y}\). This means that orthogonal matrices preserve the angles between vectors, as the dot product is related to both the magnitudes of the vectors and the cosine of the angle between them.

Preserving dot products is crucial in many applications, such as computer graphics and data analysis, where maintaining the relative orientation of data is necessary. If you think of vectors as arrows, multiplying by an orthogonal matrix rotates these arrows but does not change the angle between them.

Matrix Properties

Matrix properties, particularly when discussing orthogonal matrices, encompass a variety of unique and essential characteristics that delve into how these matrices interact with vectors and each other.

One of the most pivotal properties of orthogonal matrices is their ability to maintain orthonormality. This means that every column (or row) of matrix \(P\) is orthogonal to every other, and each column is a unit vector (vector of length 1). This is why orthogonal matrices have the condition \(P^T P = I\), ensuring that multiplying \(P\) by its transpose results in the identity matrix.

Because of these properties, orthogonal matrices enjoy several benefits:

• They have determinant values of either +1 or -1, impacting how they interact in transformations, such as reflections or rotations.
• They are inherently stable in numerical computations, reducing errors significantly when used in calculations.
• They easily preserve vector norms and dot products, as highlighted earlier, which has vast use cases in preserving distances and angles in applied mathematics.

Overall, understanding these matrix properties provides the foundation needed to grasp the larger framework of linear transformations and their applications in real-world scenarios.