Question

Let \(\mathbf{p}\) and \(\mathbf{q}\) be orthogonal idempotents. Suppose that \(\mathbf{q}=\mathbf{q}_{1}+\mathbf{q}_{2}\), where \(\mathbf{q}_{1}\) and \(\mathbf{q}_{2}\) are orthogonal idempotents. Show that \(\mathbf{q q}_{i}=\mathbf{q}_{i} \mathbf{q}=\mathbf{q}_{i}\) for \(i=1,2\). Using this result, show that \(\mathbf{p q}_{i}=\mathbf{q}_{i} \mathbf{p}=\mathbf{0}\) for \(i=1,2\).

Short answer

The expression holds true for both the idempotent and orthogonal properties by substituting the given expressions and satisfying the properties definitions.

Step-by-step solution

Prove idempotent properties

The goal is to show that \(\mathbf{q q}_{i} = \mathbf{q}_{i} \mathbf{q} = \mathbf{q}_{i}\). For starters, know that by being an idempotent, \(\mathbf{q}_{i} = \mathbf{q}_{i}^2\) . Therefore, if we multiply \(\mathbf{q}_{i}\) by \(\mathbf{q}\), \(\mathbf{q}\mathbf{q}_{i} = \mathbf{q}\mathbf{q}_{i}^2 = \mathbf{q}_{i}(\mathbf{q}\mathbf{q}_{i}) = \mathbf{q}_{i}\mathbf{q}_{i} = \mathbf{q}_{i}\(. This same logic applies to \)\mathbf{q}_{i}\mathbf{q} = \mathbf{q}_{i}\). This demonstrates the idempotent property.

Prove orthogonal properties

Now we need to show that \(\mathbf{p q}_{i}=\mathbf{q}_{i} \mathbf{p}=\mathbf{0}\) for \(i=1,2\). For orthogonal vectors, the dot product is zero. We can find this by following the same steps as above where \(\mathbf{q}_{i}=\mathbf{q}_{i} \mathbf{p}\) but instead of ending with \(\mathbf{q}_{i}\) we end with \(\mathbf{0}\(. Since \)\mathbf{p}\) is orthogonal to \(\mathbf{q}\), and \(\mathbf{q}_{i}\) is part of \(\mathbf{q}\), the result of \(\mathbf{q}_{i} \mathbf{p}\) is \(\mathbf{0}\(. Similarly, \)\mathbf{p q}_{i}=\mathbf{0}\(. Thus confirming the orthogonal property.

Key concepts

Orthogonal Idempotents

In mathematical terms, when we talk about orthogonal idempotents, we're interested in particular types of matrices or elements in algebraic structures. Idempotent elements are those that, when multiplied by themselves, yield the same element. For instance, if we have an idempotent element denoted as \(\mathbf{e}\), the idempotency property can be expressed as \(\mathbf{e}^2 = \mathbf{e}\). This characteristic is fascinating because it creates a kind of stability within the system—once you reach an idempotent element, repeating the operation doesn't change anything.

Now, when we say two idempotent elements are orthogonal, we're adding an additional layer. This means not only do \(\mathbf{e}\) and another idempotent \(\mathbf{f}\) satisfy their respective idempotent property, but their product is also zero. Symbolically, if \(\mathbf{e}\) and \(\mathbf{f}\) are orthogonal idempotents, then \(\mathbf{e}\mathbf{f} = \mathbf{f}\mathbf{e} = \mathbf{0}\).

Why is this concept relevant? In linear algebra and its applications, orthogonal idempotents help decompose spaces and solve equations efficiently. They have applications in projection operations, and understanding them helps in breaking down more complex problems into simpler parts.

Idempotent Matrices

Moving to the concept of idempotent matrices, we find parallels with the property discussed before. An idempotent matrix is one that, when multiplied by itself, results in the same matrix. The matrix form of the idempotency equation is \(\mathbf{A}^2 = \mathbf{A}\). These matrices are immensely useful as they represent projection operations in linear algebra—operations mapping a vector space onto itself in a manner that repeatedly applying the operation does not change the result.

The computation process for proving matrix idempotency might involve verifying that \(\mathbf{A}^2\) indeed equals \(\mathbf{A}\). This can sometimes seem tedious but is crucial to establish the properties of the matrix in question. In applications such as statistics, idempotent matrices can emerge in the context of linear regression models, where they help in deriving estimators with desirable properties.

Understanding the idempotent matrices lays the groundwork for grasping more complex linear algebra concepts, enabling students to approach problems analytically and with confidence.

Orthogonal Vectors

Finally, we turn our attention to orthogonal vectors. In the context of vectors, orthogonality means perpendicularity. When two vectors are orthogonal, their dot product equals zero: \(\mathbf{v} \cdot \mathbf{w} = 0\). This zero result intuitively means that there is no projection of one vector onto the other — they are aligned at right angles to each other.

In higher dimensions, visualizing orthogonality can be challenging, but the principle remains the same. Orthogonal vectors form the basis of many geometrical and algebraic constructs. For instance, in three-dimensional space, the standard basis vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are mutually orthogonal, simplifying the process of breaking down vectors into components and understanding the structure of the space.

In practice, finding orthogonal vectors is a routine part of many procedures in mathematics, physics, and engineering. For learners, mastering the concept helps in solving for equilibrium conditions, analyzing forces, and optimizing designs. Another benefit is in numerical methods, where orthogonal vectors are part of strategies to minimize error and improve computational stability.