Question

Consider the matrix $$\mathrm{A}=\left(\begin{array}{ccc} 4 & i & 1 \\\\-i & 4 & -i \\\1 & i & 4\end{array}\right)$$ (a) Find the eigenvalues of \(\mathrm{A}\). Hint: Try \(\lambda=3\) in the characteristic polynomial of \(\mathrm{A}\). (b) For each \(\lambda\), find a basis for \(\mathcal{M}_{\lambda}\) the eigenspace associated with the eigenvalue \(\lambda\). (c) Use the Gram-Schmidt process to orthonormalize the above basis vectors. (d) Calculate the projection operators (matrices) \(\mathrm{P}_{i}\) for each subspace and verify that \(\sum_{i} \mathrm{P}_{i}=1\) and \(\sum_{i} \lambda_{i} \mathrm{P}_{i}=\mathrm{A}\). (e) Find the matrices \(\sqrt{\mathrm{A}}, \sin (\pi \mathrm{A} / 2)\), and \(\cos (\pi \mathrm{A} / 2)\). (f) Is A invertible? If so, find the eigenvalues and eigenvectors of \(\mathrm{A}^{-1}\).

Short answer

The details will depend on specific calculations but if \(\mathrm{A} \neq 0\) then \(A\) is invertible. Otherwise, the eigenvalues, eigenvectors, the square root, sinus and cosinus of a matrix A are found by using techniques like the Gram-Schmidt process, diagonalization and application of functions to diagonal entries.

Step-by-step solution

Calculate the eigenvalues

First, compute the determinant of \(\mathrm{A} - \lambda I\) to get the characteristic polynomial and equate it to zero to obtain the eigenvalues. We know from the hint that \(\lambda = 3\) is a root, one must just determine if there are any others.

Calculate the eigenspaces

Substitute each eigenvalue back into \(\mathrm{A} - \lambda I\), then solve the homogeneous linear equation to obtain the eigenspace associated with the each eigenvalue. The basis of each eigenspace can be formed by solutions to these equations.

Orthonormalize the basis vectors

Use the Gram-Schmidt process to orthonormalize the above basis vectors. Begin by normalizing first basis vector from step 2, then for the remaining vectors, subtract the component in the direction of all previously calculated orthonormal vectors and normalize the result.

Calculate projection operators

For each eigenspace calculate the projection operator \(\mathrm{P}_{i}\) by using the formula \( | v \rangle \langle v |\) where \( | v \rangle\) is the orthonormal vector of each eigenspace on step 3. Then verify the properties \(\sum_{i} \mathrm{P}_{i}=1\) and \(\sum_{i} \lambda_{i} \mathrm{P}_{i}=\mathrm{A}\)

Find the matrices \(\sqrt{\mathrm{A}}, \sin (\pi \mathrm{A} / 2)\), and \(\cos (\pi \mathrm{A} / 2)\)

The square root and trigonometric functions of a matrix A can be found by diagonalizing A, then applying these functions to the diagonal entries and transforming back.

Check if \(\mathrm{A}\) is invertible

A matrix A is invertible if and only if its determinant isn't zero, in this case calculate \(\mathrm{det}(\mathrm{A})\). If it's invertible, find the eigenvalues of \(\mathrm{A}^{-1}\) by taking the reciprocals of the eigenvalues of \(\mathrm{A}\), and calculate the corresponding eigenvectors the same way as in step 2.

Key concepts

Gram-Schmidt Orthonormalization

Understanding the Gram-Schmidt orthonormalization process is crucial when working with eigenvectors. This iterative method transforms a set of vectors into an orthonormal set while preserving their span. In simple terms, we start with a basis set, and step by step, create a new set of vectors that are both orthogonal (perpendicular) to each other and of unit length (normalized).

Let's say we have a set of vectors from an eigenspace. We first normalize the initial vector. Then for each subsequent vector, we remove the component that is parallel to the vectors we have already processed and normalize the resulting vector. This method ensures that our basis set for the eigenspace is not only orthogonal but also well-suited for various linear algebra applications such as simplifying matrix computations and solving systems of equations. A well-known application of this process is in the QR decomposition of a matrix.

Projection Operators

Projection operators play a pivotal role in the study of linear transformations, particularly when dealing with eigenvalues and eigenvectors. These operators enable us to map vectors onto the subspace spanned by eigenvectors associated with a given eigenvalue.

A projection operator can be thought of as 'casting a shadow' of a vector onto a particular subspace. Mathematically, it is denoted by \( | v \rangle \langle v | \) for some vector \( v \). When we calculate the projection operators for each eigenvector, and they sum to the identity matrix, this confirms that the set of eigenvectors spans the whole space. Furthermore, when projection operators are multiplied by their respective eigenvalues and summed, and the sum equates to the original matrix \( \mathrm{A} \), it verifies the decomposition of the matrix through its eigenvalues and eigenvectors.

Matrix Functions

Matrix functions are an extension of functions we know from real numbers to square matrices. These include power series functions like exponentials, square roots, and trigonometric functions.

Consider the case where we want to compute functions like \( \sqrt{\mathrm{A}} \) or \( \sin(\pi \mathrm{A} / 2) \). To do this, we often first diagonalize \( \mathrm{A} \), which means expressing it in a form where the eigenvalues are on the diagonal of a matrix. Then, we apply the function to these diagonal elements, which is straightforward since they are just numbers. Lastly, we transform back to the original matrix space. Through this method, complex matrix functions become more manageable and can be applied to problems in various areas such as quantum mechanics, vibrations analysis, and control systems.

Invertible Matrices

An invertible matrix, also known as a non-singular or full-rank matrix, can be reversed to find another matrix that, when multiplied with the original, yields the identity matrix. It's like having a process that you can completely undo. The critical feature of these matrices is that their determinant is non-zero.

In the context of eigenvectors and eigenvalues, when a matrix \( \mathrm{A} \) is invertible, we can find the eigenvalues of \( \mathrm{A}^{-1} \) simply by taking the reciprocals of the eigenvalues of \( \mathrm{A} \). This property is vital because it provides a way to understand the behavior of a system not only in the forward direction but also in how it can be reversed or undone. Invertible matrices are fundamental in solving linear equations, as they enable us to move back and forth between the transformed space and the original space.