A simple demonstration of the existence of the jordan canonical form for any square matrix

All the demonstrations known to this author of the existence of the Jordan Canonical Form are somewhat complex - usually invoking the use of new spaces, and what not. These demonstrations are usually too difficult for an average Mathematics student to understand how he or she can obtain the Jordan Canonical Form for any square matrix. The method here proposed not only demonstrates the existence of such forms but, additionally, shows how to find them in a step by step manner. I do not claim that the following demonstration is in any way “elegant” (by the standards of elegance in fashion nowadays among mathematicians) but merely simple (undergraduate students taking a fist course in Matrix Algebra would understand how it works).

A monthly indicator of Brazilian GDP. The Brazilian business cycle and growth cycle

This paper constructs an indicator of Brazilian GDP at the monthly ftequency. The peculiar instability and abrupt changes of regimes in the dynamic behavior of the Brazilian business cycle were explicitly modeled within nonlinear ftameworks. In particular, a Markov switching dynarnic factor model was used to combine several macroeconomic variables that display simultaneous comovements with aggregate economic activity. The model generates as output a monthly indicator of the Brazilian GDP and real time probabilities of the current phase of the Brazilian business cycle. The monthly indicator shows a remarkable historical conformity with cyclical movements of GDP. In addition, the estimated filtered probabilities predict ali recessions in sample and out-of-sample. The ability of the indicator in linear forecasting growth rates of GDP is also examined. The estimated indicator displays a better in-sample and out-of-sample predictive performance in forecasting growth rates of real GDP, compared to a linear autoregressive model for GDP. These results suggest that the estimated monthly indicator can be used to forecast GDP and to monitor the state of the Brazilian economy in real time.

The Brazilian foreign exchange market through the microstructure perspective

The objective of this study is to investigate whether the relationship between order ow and the spot exchange rate stems from the fact that the ow aggregates information on dispersed economic fundamentals in the economy. To perform this test, a database that includes all transactions of the commercial and nancial segments of the Brazilian primary foreign exchange market between January of 1999 and May of 2008 was used. We show that the order ow was partly responsible for variations in in ation expectations over the time period and that this relationship did not remain robust, drawing comparisons with other fundamentals such as GDP and Industrial Production.