# Browse Classroom Capsules and Notes

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Displaying 61 - 70 of 85

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.