# Monthly Archives: June 2011

## Linear Algebra and Its Applications, Exercise 1.6.20

Exercise 1.6.20. Suppose is a 3 by 3 matrix for which the third row is the sum of the first and second rows. Show that there is no solution to the equation , and determine whether has an inverse or … Continue reading

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## Linear Algebra and Its Applications, Exercise 1.6.19

Exercise 1.6.19. Given the matrix where what values must the entries of have in order for to be invertible? Answer: We know that a (square) matrix is invertible if and only if it is nonsingular. We therefore perform Gaussian elimination … Continue reading

## The inverse of a block diagonal matrix

In the previous post I discussed multiplying block diagonal matrices as part of my series on defining block diagonal matrices and partitioning arbitrary square matrices uniquely and maximally into block diagonal form (part 1, part 2, part 3, part 4, … Continue reading

## Multiplying block diagonal matrices

In a previous post I discussed the general problem of multiplying block matrices (i.e., matrices partitioned into multiple submatrices). I then discussed block diagonal matrices (i.e., block matrices in which the off-diagonal submatrices are zero) and in a multipart series … Continue reading

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## Partitioning a matrix into block diagonal form, part 5

In previous posts in this series I informally explored how to partition a matrix into block diagonal form (part 1), described a method for constructing a partitioning vector for an by square matrix (part 2), showed that the vector thus … Continue reading

## Partitioning a matrix into block diagonal form, part 4

In my previous two posts I described a method for constructing a partitioning vector for an by square matrix (part 2 of this series) and showed that the vector thus constructed partitions into block diagonal form (part 3). In this … Continue reading

## Partitioning a matrix into block diagonal form, part 3

In part 2 of this series I described a method to construct a vector that partitions an by square matrix into row and column partitions. In this post I show that the vector thus constructed partitions into block diagonal form. … Continue reading

## Partitioning a matrix into block diagonal form, part 2

In part 1 of this series I outlined an informal method to partition a square matrix into block diagonal form. In this post I provide a formal description of such a method, applicable to any by square matrix. In doing … Continue reading

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## Partitioning a matrix into block diagonal form, part 1

In my previous post I offered a more formal definition of a block diagonal matrix, and claimed that we can partition an arbitrary by square matrix into block diagonal form in a way that is both maximal (i.e., no partitioning … Continue reading