eigenvalue problems – Linear Combinations

Representing a Quadratic Form Using a Matrix

March 18, 2021October 10, 2021greg.mayer

To motivate the use of linear algebra to study quadratic forms, consider the following problem. Does this inequality hold for all real values of $x$ and $y$? \begin{equation*} 5x^2 + 8 y ^2 – 4 xy \geq 0 \end{equation*} Were…Continue readingRepresenting a Quadratic Form Using a Matrix

Eigenvalues of Real Symmetric Matrices

February 27, 2021October 10, 2021greg.mayer

The eigenvalues of a square matrix with real entries can be complex. For example, a quick calculuation will verify that the eigenvalues of $$A = \begin{pmatrix} 1&-2\\2&1\end{pmatrix}$$ are $1\pm2i$. But when the matrix is symmetric, the eigenvalues must be real.…Continue readingEigenvalues of Real Symmetric Matrices

Why are Eigenvectors from Distinct Eigenvalues Linearly Independent?

December 9, 2020January 8, 2021greg.mayer

In this post we will give a proof of the following theorem. Theorem If $\vec v_1, \vec v_2, \ldots , \vec v_r$ are eigenvectors of matrix $A$ that correspond to distinct eigenvalues $\lambda_1, \lambda_2, \ldots , \lambda_r$, then the set…Continue readingWhy are Eigenvectors from Distinct Eigenvalues Linearly Independent?

Diagonalizability and Invertibility of a Matrix

July 16, 2020January 8, 2021greg.mayer

Is there a relationship between invertibility of a matrix and whether it can be diagonalized? For example, if a matrix is not invertible, can the matrix still be diagonalized? Before we explore these relationships, let’s give definitions for the invertibility…Continue readingDiagonalizability and Invertibility of a Matrix

Seven Different Ways to Obtain the Eigenvalues of a Matrix

February 19, 2020January 8, 2021greg.mayer

When asked to compute the eigenvalues of a real $n \times n$ matrix $A$, students in an introductory linear algebra course are often taught that they can calculate the roots of the polynomial $\text{det} (A – \lambda I)$. The roots…Continue readingSeven Different Ways to Obtain the Eigenvalues of a Matrix