Linear Algebra and Its Applications, Review Exercise 2.19

Review exercise 2.19. Consider the set of elementary 3 by 3 matrices $E_{ij}$ with ones on the diagonal and at most one nonzero entry below the diagonal. What subspace is spanned by these matrices?

Answer: An example member of this set is

$E_{32} = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&1&1 \end{bmatrix}$

Since each matrix in the set has only zeros above the diagonal, a linear combination pf such matrices will also have only zeros above the diagonal.

Since each matrix in the set has ones on the diagonal, multiplying a given matrix by a scalar produces a matrix which has that scalar value for all diagonal entries. Adding a number of such matrices in turn produces a matrix for which all diagonal entries are equal.

Since each matrix in the set has some nonzero entry below the diagonal in some arbitrary location, a linear combination of such matrices could have any arbitrary set of values below the diagonal.

The subspace spanned by the set of 3 by 3 elementary matrices $E_{ij}$ is therefore the set of all 3 by 3 lower triangular matrices for which the diagonal values are equal to one another.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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