Category Archives: linear algebra

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 →

This and subsequent posts continue the discussion in my previous post about multiplying block matrices. In this series of posts I consider block diagonal matrices and in particular how to partition an existing matrix into block diagonal form. I’m doing … Continue reading →

Exercise 1.6.18. Suppose that Show that Answer: We have The product of two upper triangular matrices is also an upper triangular matrix, and multiplying by a diagonal matrix preserves this. The left side of the final equation above is therefore … Continue reading →

Exercise 1.6.16. (i) If A is an n by n symmetric matrix, how many entries of A can be chosen independently of each other? (ii) If If K is an n by n skew-symmetric matrix, how many entries of K … Continue reading →

Exercise 1.6.15. For any square matrix B and matrices A and K where prove that A is symmetric and K is skew-symmetric, i.e., For the case where compute A and K and show that B can be expressed as the … Continue reading →

Exercise 1.6.13. Compute , , , and for the following matrices: Answer: We have Note that per Equation 1M(i) on page 47 we have This means that and as confirmed by the computation above. NOTE: This continues a series of … Continue reading →

Exercise 1.6.12. Suppose that A is an invertible matrix and has one of the following properties: (1) A is a triangular matrix. (2) A is a symmetric matrix. (3) A is a tridiagonal matrix. (4) All the entries of A … Continue reading →

Exercise 1.6.11. For each of the following criteria, give examples of matrices A and B that satisfy the criteria: (i) Both A and B are invertible, but their sum A+B is not invertible. (ii) Neither A nor B is invertible, … Continue reading →

In doing exercise 1.6.10 in Linear Algebra and Its Applications I was reminded of the general issue of multiplying block matrices, including diagonal block matrices. This also came up in exercise 1.4.24 as well, which I answered without necessarily fully … Continue reading →

Exercise 1.6.10. Determine the inverses of the following matrices: Answer: The first matrix has the form of a diagonal matrix, only flipped horizontally, and it’s therefore worth seeing if its inverse has an analogous form to the inverse of a … Continue reading →