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Category Archives: matrices
Determinant of a Symmetric Pascal Matrix
The infinite symmetric Pascal matrix Q is given by where entry in Q is . (Note that we begin indexing the matrix with 0, not 1, in keeping with the way Pascal’s triangle is usually indexed.) The purpose of this post is to … Continue reading
Posted in binomial coefficients, matrices
2 Comments
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
Posted in Fibonacci sequence, matrices
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Symmetric 01 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 01 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 nonnegative and whose rows all sum to 1. The transition matrix for a finitestate 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 01 Matrices with All Eigenvalues Positive
Recently I was surprised to learn that the only symmetric 01 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 01 matrix … Continue reading
Posted in linear algebra, matrices
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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