Eigenvalues and Eigenvectors
https://www.maa.org/taxonomy/term/42306/all
enThe Axis of a Rotation: Analysis, Algebra, Geometry
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-axis-of-a-rotation-analysis-algebra-geometry
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Given any 3 by 3 rotation matrix \(A\) (i.e. orthogonal with determinant \(1\) and an arbitrary vector \(x\), the vector \( Ax +A^{T}x+[1−\)tr\((A)]x\) lies on the axis of rotation. The article provides three different approaches, requiring various levels of background knowledge, to prove and/or explain the given result.</em></p>
</div></div></div>Using Quadratic Forms to Correct Orientation Errors in Tracking
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/using-quadratic-forms-to-correct-orientation-errors-in-tracking
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>If noise in data transmission produces a not quite orthogonal matrix that is known to be orthogonal, how does one find the "nearest" orthogonal matrix? This capsule recasts the problem as one of maximizing a quadratic form on the four-dimensional unit sphere, and sketches a solution.</em></p>
</div></div></div>Finding the Volume of an Ellipsoid Using Cross-Sectional Slices
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/finding-the-volume-of-an-ellipsoid-using-cross-sectional-slices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author shows that the volume of an ellipsoid can be determined by three parallel slices.</em></p>
</div></div></div>Matrices, Continued Fractions, and Some Early History of Iteration Theory
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrices-continued-fractions-and-some-early-history-of-iteration-theory
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Continued fractions of the form \( \frac{1}{1 + \frac{c}{1 + \frac{c}{ 1 +\ddots}}} \) are analyzed using linear algebra and iteration theory. The continued fractions of interest are closely related to a class of \(2 \times 2\) matrices, and the eigenvalues and eigenvectors of those matrices are investigated to determine when the corresponding continued fractions converge. Historical references are included.</em></p>
</div></div></div>Characteristic Polynomials of Magic Squares
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/characteristic-polynomials-of-magic-squares
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>An \(n \times n \) matrix whose rows, columns, and diagonal all sum to the same number \(m\) is called magic, and the number \(m\) is called the magic sum. If \(A\) is a magic square matrix, then its magic sum \(m\) must be an eigenvalue, and hence a characteristic root, of \(A\). A main result of this paper shows that the sum of all the characteristic roots of \(A\) except for \(m\) must be zero.</p>
</div></div></div>Determinants of Sums
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinants-of-sums
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>An interesting formula for the determinant of the sum of any two matrices of the same size is presented. The formula can be used to obtain important results about the characteristic polynomial and about the characteristic roots and subdeterminants of the matrices being added.</em></p>
</div></div></div>Complex Eigenvalues and Rotations: Are Your Students Going in Circles?
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/complex-eigenvalues-and-rotations-are-your-students-going-in-circles
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author shows that every \(2 \times 2\) real matrix with nonreal eigenvalues represents the composition of the following three operations: (1) a vertical “lift” to a plane through the origin, (2) a rotation in that plane, and (3) a “drop” back into the \(x-y\)-plane.</em></p>
</div></div></div>Eigenvalues of Matrices of Low Rank
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/eigenvalues-of-matrices-of-low-rank
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A technique is discussed for finding the eigenvalues of square matrices of small rank, which is useful for student discovery in a linear algebra class. The eigenvalues of a matrix of rank 1 or 2 can be found by solving a linear or quadratic equation.</em></p>
</div></div></div>Collapsed Matrices with (almost) the Same Eigenstuff
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/collapsed-matrices-with-almost-the-same-eigenstuff
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author describes a method for constructing a smaller matrix with the same (or similar) eigenvalues that would be usable in the classroom. He illustrates this with matrices for Leslie population models.</em></p>
</div></div></div>Clock Hands Pictures for \(2 \times 2\) Real Matrices
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/clock-hands-pictures-for-2-times-2-real-matrices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The article illustrates the notion of eigenvalue and its corresponding eigenvector using hands of an analog clock. This capsule deals with \(2 \times 2\) real matrices, single eigenvalues, and complex ones as well.</em></p>
</div></div></div>The Arithmetic of Algebraic Numbers: An Elementary Approach
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-arithmetic-of-algebraic-numbers-an-elementary-approach
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>If \(r\) and \(s\) are algebraic numbers, then \(r + s\), \(rs\), and \(r/s\) are also algebraic. The proof provided in this capsule uses the ideas of characteristic polynomials, eigenvalues, and eigenvectors.</em></p>
</div></div></div>Facial Recognition
https://www.maa.org/programs/faculty-and-departments/course-communities/facial-recognition
Linear Algebra Demos, Drexel Univ
https://www.maa.org/programs/faculty-and-departments/course-communities/linear-algebra-demos-drexel-univ
Conceptual Linear Algebra Online
https://www.maa.org/programs/faculty-and-departments/course-communities/conceptual-linear-algebra-online
Online Math Lab at UC Santa Barbara
https://www.maa.org/programs/faculty-and-departments/course-communities/online-math-lab-at-uc-santa-barbara