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Browse Classroom Capsules and Notes

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The author presents a method to transfer matrix identities over the real numbers to matrix identities over an arbitrary commutative ring.  Several examples are given, including \(\det(AB)= \...

One version of the Principal Axis Theorem asserts that any symmetric matrix with entries in \( \mathcal{R}\) is similar over \(\mathcal{R}\) to a diagonal matrix.  The authors find...

Let \(C_f\) be the companion matrix of a polynomial \(f\) with coefficients in an integral domain \(D\).  Let \(Q\) be the quotient field of \(D\).  By use of the Jordan form of \(...

Not all square matrices commute with their generalized inverse (Moore-Penrose inverse). The author gathers equivalent conditions for the generalized inverse of a matrix to commute with the...

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...

Two least-squares methods are used to estimate city and highway gas mileage from readily measured data. First, the author uses the standard least squares method which is suitable for a first...

Newton’s identities relate the coefficients of a polynomial to sums of powers of its roots.  The author uses the Cayley-Hamilton theorem and properties of the trace of a matrix to...

A short elementary proof of the equality of row rank and column rank is given. The proof requires only the definition of matrix multiplication and the fact that a minimal spanning set is a...

Consider the vector space of polynomials of degree less than \(n\), and a polynomial \(p(x)\) in this space. The author describes the matrix \(M(r) \) that maps the polynomial \(p(x)\) to \(p(...

Using mathematical induction, the author provides a simple proof that the reduced row echelon form of a matrix is unique.

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