Question

Find the eigenvalue(s) and eigenvector(s) for the following: a. \(\left(\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right)\) b. \(\left(\begin{array}{ll}3 & -5 \\ 1 & -1\end{array}\right)\) c. \(\left(\begin{array}{ll}4 & 1 \\ 0 & 4\end{array}\right)\) d. \(\left(\begin{array}{ccc}1 & -1 & 4 \\ 3 & 2 & -1 \\ 2 & 1 & -1\end{array}\right)\)

Short answer

a. Eigenvalues are \(λ = 1, 6\) and their corresponding eigenvectors are \([1, 1]\) and \([-1, 1]\). b. Eigenvalues are \(λ = -2, 4\) and their corresponding eigenvectors are \([-5, 1]\) and \([1, 1]\). c. Eigenvalue is \(λ = 4\) with algebraic multiplicity of 2, and the corresponding eigenvectors are \([1, 0]\) and \([0, 1]\). d. Eigenvalues are \(λ = 4, -2\) and their corresponding eigenvectors are \([1, 1, 1]\) and \([-2, 1, 0]\).

Step-by-step solution

Find the Eigenvalues

Compute the characteristic equation by subtracting \(λ\) from the diagonals of the matrix and calculating the determinant, set equal to zero. Then, solve this equation for \(λ\). This step is done for each given matrix.

Find the Eigenvectors

After finding the eigenvalues, substitute them back into the equation (Matrix - λ * Identity Matrix) = 0 to get a system of homogeneous linear equations. Then, solve the systems of equations to find the eigenvectors. Repeat these steps for each matrix.

Simplify the Eigenvector

Each resulting solution vector from Step 2 is an eigenvector. Make sure to simplify it if necessary. In some cases, there could be multiple (linearly independent) eigenvectors for a given eigenvalue.

Key concepts

Characteristic Equation

The characteristic equation is a foundational concept when identifying both the eigenvalues and eigenvectors of a matrix. It is a polynomial equation that emerges when we subtract a scalar multiple of the identity matrix from our square matrix and then calculate the determinant of the resulting matrix.

To get a clearer picture of this, consider a square matrix 'A'. To locate its eigenvalues, we shift our focus on the equation \(\text{det}(A - \lambda I) = 0\) where \(\lambda\) is a scalar known as the eigenvalue, and 'I' is the identity matrix of the same dimensions as 'A'. Solving the equation \(\text{det}(A - \lambda I) = 0\), or the characteristic equation, will give us the values of \(\lambda\) for which the equation holds true. These values are crucial as they directly relate to the behavior and properties of the matrix.

Homogeneous Linear Equations

Homogeneous linear equations play a pivotal role in finding eigenvectors after the eigenvalues have been determined. A homogeneous linear equation is one where the result is set to zero: \(Ax = 0\). In the scenario of eigenvectors, given a matrix 'A' and its eigenvalue \(\lambda\), we examine the system \(\left(A - \lambda I\right)x = 0\).

This system of equations, which has zero on the right side, implies that we are looking for non-trivial solutions for the vector 'x' that satisfy the equation. Solving this yields one or more vectors, and these solutions are the eigenvectors we're seeking. They demonstrate the direction in which the transformation represented by 'A' stretches or compresses but does not change direction. The homogeneity of these equations implies that if 'x' is an eigenvector, then any scalar multiple of 'x' is also an eigenvector.

Identity Matrix

The identity matrix, typically denoted by 'I', is the multiplicative neutral element of the square matrix world. It plays a functional role in various matrix computations, acting like the number '1' in scalar arithmetic. In every identity matrix, the diagonal elements are '1', and all other elements are '0'.

When discussing eigenvalues and eigenvectors, the identity matrix is integral to the formation of the characteristic equation. By subtracting \(\lambda\) times the identity matrix from our matrix 'A', we're effectively shifting the eigenvalues of 'A' down the diagonal, setting the stage for our determinant calculation that ultimately leads us to the characteristic equation \(\text{det}(A - \lambda I) = 0\). The identify matrix's structure ensures that this subtraction doesn't alter the locations of the original matrix's off-diagonal entries, focusing the operation purely on its eigenvalues.