Category Archives: matrices

Pascal Matrices and Binomial Inversion

In this post we’ll look at the relationship between a Pascal matrix, its inverse, and binomial inversion.  It turns out that these are the same concepts viewed from two different angles. The Pascal matrix is the matrix containing Pascal’s triangle … Continue reading

Posted in binomial coefficients, matrices | 2 Comments

Cassini’s Identity without Matrix Powers

Cassini’s identity for Fibonacci numbers says that .  The classic proof of this shows (by induction) that .  Since , Cassini’s identity follows. In this post I’m going to give a different proof involving determinants, but one that does not use … Continue reading

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Symmetric 0-1 Matrices with All Eigenvalues Positive, Part 2

A recent post and question on math.SE ask for an intuitive proof of the fact that the identity is the only symmetric 0-1 matrix with all eigenvalues positive.  As I mentioned in that recent post, Robert Israel’s argument is quite nice, but … Continue reading

Posted in linear algebra, matrices | 4 Comments

A Simple Proof that the Largest Eigenvalue of a Stochastic Matrix is 1

A stochastic matrix is a square matrix whose entries are non-negative and whose rows all sum to 1.  The transition matrix for a finite-state Markov chain is a stochastic matrix, and so they are essential for tackling problems that can … Continue reading

Posted in linear algebra, matrices, probability | 12 Comments

Symmetric 0-1 Matrices with All Eigenvalues Positive

Recently I was surprised to learn that the only symmetric 0-1 matrix with all eigenvalues positive is the identity matrix.  Here’s a very nice proof of this fact given in this answer of Robert Israel’s: Let A be an symmetric 0-1 matrix … Continue reading

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When Do 2×2 Covariance Matrices Have Repeated Eigenvalues?

The calculation answering the question in the title is not that difficult, but I needed it recently, and so I thought I would add it to my blog. Suppose we have a covariance matrix By definition, the eigenvalues are the … Continue reading

Posted in linear algebra, matrices | 2 Comments