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The characteristic equation of a matrix A is defined by

det| A - λI | = 0 

where λ is a complex (scalar) parameter, I is the identity matrix and det|...| is the determinant. The left-hand side evaluates to a polynomial in λ, the characteristic polynomial, which is quadratic in the case of 2x2 matrices. The solutions of this characteristic equation are the eigenvalues of A, which we will denote as λ₁ and λ₂.

Now the eigenvectors v_i of A satisfy

A vi = λi vi

For each λ_i, this gives you a system of two equations in two unknowns (the components of v_i), which can be solved quite easily. You will notice that the system is actually underspecified, and the magnitude of the eigenvectors are not determined by the equations. We will usually want the eigenvectors to be normalised, that is √(|x|² + |y|²) = 1, where x and y are the vector components, |x|² is x multiplied by its complex conjugate.

Note that the eigenvalues may be degenerate, i.e. λ₁ = λ₂. In this case, you may or may not be able to satisfy the single system of equations with two linearly independent eigenvectors.

Given a 2x2 matrix with complex elements, determine its two (possibly identical) eigenvalues and a normalised eigenvector for each eigenvalue. The resulting numbers must be accurate to at least 3 (decimal) significant digits. You may assume that the real and imaginary parts of any matrix element is in the range [-1,1].

The characteristic equation of a matrix \$A\$ is defined by

$$\det| A - λI | = 0$$

where \$λ\$ is a complex (scalar) parameter, \$I\$ is the identity matrix and \$\det|...|\$ is the determinant. The left-hand side evaluates to a polynomial in \$λ\$, the characteristic polynomial, which is quadratic in the case of 2x2 matrices. The solutions of this characteristic equation are the eigenvalues of \$A\$, which we will denote as \$λ_1\$ and \$λ_2\$.

Now the eigenvectors \$v_i\$ of \$A\$ satisfy

$$Av_i = λ_iv_i$$

For each \$λ_i\$, this gives you a system of two equations in two unknowns (the components of \$v_i\$), which can be solved quite easily. You will notice that the system is actually underspecified, and the magnitude of the eigenvectors are not determined by the equations. We will usually want the eigenvectors to be normalised, that is \$\sqrt{|x|^2 + |y|^2} = 1\$, where \$x\$ and \$y\$ are the vector components, \$|x|^2\$ is \$x\$ multiplied by its complex conjugate.

Note that the eigenvalues may be degenerate, i.e. \$λ_1 = λ_2\$. In this case, you may or may not be able to satisfy the single system of equations with two linearly independent eigenvectors.

Given a 2x2 matrix with complex elements, determine its two (possibly identical) eigenvalues and a normalised eigenvector for each eigenvalue. The resulting numbers must be accurate to at least 3 (decimal) significant digits. You may assume that the real and imaginary parts of any matrix element is in the range \$[-1,1]\$.

For each λ_i, this gives you a system of two equations in two unknowns (the components of v_i), which can be solved quite easily. You will notice that the system is actually underspecified, and the magnitude of the eigenvectors are not determined by the equations. We will usually want the eigenvectors to be normalised, that is √(|x|² + |y|²) = 1, where x and y are the vector components.

For each λ_i, this gives you a system of two equations in two unknowns (the components of v_i), which can be solved quite easily. You will notice that the system is actually underspecified, and the magnitude of the eigenvectors are not determined by the equations. We will usually want the eigenvectors to be normalised, that is √(|x|² + |y|²) = 1, where x and y are the vector components, |x|² is x multiplied by its complex conjugate.