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Browse Classroom Capsules and Notes

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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...

Suppose we have several alternatives that we wish to rank.  For example, we may wish to rank five teachers according to their teaching excellence.  The author constructs a positive...

The author discusses the relationship of two well-known, apparently unrelated supermultiplicative functions of nonnegative matrices, and shows they are special cases of a more general...

The author derives formulas for the coefficients of the characteristic polynomial of a square matrix in terms of the partial derivatives of its determinant, which is viewed as a function of...

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...

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...

Consider \(n\) vectors issuing from the origin in \(n\)-dimensional space.  The author shows that the statement “any set of \(n\) vectors in \(n\)-space, no two of which meet at greater...

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)= \...

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