Let A=[ a b c d ] be a real matrix. Find...

Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ be a real matrix. Find necessary and sufficicnt conditions on $a, b, c, d$ so that $A$ is diagonalizable - that is, so that $A$ has two (real) linearly independent eigenvectors.

Key Concepts

Diagonalizability

This refers to the ability of a matrix to be expressed in a basis consisting entirely of its eigenvectors. For a 2x2 matrix, it means that there exist two linearly independent eigenvectors so that the matrix is similar to a diagonal matrix. Diagonalizability simplifies many computations, including raising matrices to powers, and is closely tied to the eigenstructure of the matrix.

Eigenvalues

Eigenvalues are scalars associated with a matrix such that when the matrix acts on its eigenvector, the product is the eigenvector scaled by the eigenvalue. In the context of diagonalizability, having distinct eigenvalues (which occur when the characteristic polynomial has two distinct real roots) guarantees the existence of a basis formed by eigenvectors.

Characteristic Polynomial

The characteristic polynomial is a polynomial derived from a matrix and is used to determine its eigenvalues. For a 2x2 matrix, it is a quadratic polynomial whose roots yield the eigenvalues. Its form and the nature of its roots (real, repeated, or complex) provide important information about the matrix's diagonalizability.

Discriminant

The discriminant of the characteristic polynomial is the part of the quadratic formula under the square root. It indicates whether the eigenvalues are distinct, repeated, or complex. For a 2x2 matrix to be diagonalizable over the reals, the discriminant must be positive so that the eigenvalues are distinct and real, ensuring the existence of two independent eigenvectors.

Transcript

Please give Ace some feedback