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Tomorrow I've got a test on Linear Algebra and I've come across elementary matrices $U_{i,j}(a)=I_m +aE_{i,j}$, $V_{i,j}= I_m + E_{i,j} + E_{j,i}-E_{i,i}-E_{j,j}$ and $W_i(c)= I_m + (c-1)E_{i,i}$

I want to know what are the applications of these elementary matrices. I've already noticed you can use a combination of the elementary matrices multiplied by a matrix A to get the $ref(A)$.

But are there any other uses for these special kind of matrices.

EDIT: There are some identities of these elementary matrices like for instance $U_{i,j}(a)\cdot U_{i,j}(-a)= I_m$, $V_{i,j}\cdot V_{i,j}= I_m$, $W_i(c)\cdot W_i(c^{-1})=I_m$. Do you know some other special identities using these elementary matrices.

Anonymous196
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  • They come into algebraic K-theory. – Angina Seng Oct 26 '17 at 18:34
  • What does that exactly entail? – Anonymous196 Oct 26 '17 at 18:35
  • My experience is that at an intro level they are only used as a device to prove things about determinants. – Randall Oct 26 '17 at 18:36
  • Elementary matrices are precisely the matrices that correspond to elementary row operations in Gaussian and Gauss-Jordan elimination. –  Oct 26 '17 at 18:40
  • There are other uses in more advanced mathematics but they probably won't come up on your exam. If there is a question using them it will be a straightforward test to see if you understand them. You're right that their main value at your level is showing how the row operations in Gaussian elimination (finding rref) can be written using matrix multiplication. – Ethan Bolker Oct 26 '17 at 18:41
  • I somehow wonder there is much more to these type of matrices because of all the special identities that follow – Anonymous196 Oct 26 '17 at 18:44

2 Answers2

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Here is an example. Given a square symmetric matrix $H,$ we can use elementary matrices to perform one step at a time to construct $P^T H P = D,$ where $D$ is diagonal and $\det P = \pm 1.$ As the inverse of an elementary matrix is another (evident) elementary matrix, we can also use these to construct $Q = P^{-1}$ a step at a time.

What follows below is the way I like to display the algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr

$$ P^T H P = D $$ $$ Q^T D Q = H $$ $$ H = \left( \begin{array}{rrr} 0 & 2 & 3 \\ 2 & 1 & 5 \\ 3 & 5 & 10 \\ \end{array} \right) $$

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$$\left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P = \left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q = \left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D = \left( \begin{array}{rrr} 1 & 2 & 5 \\ 2 & 0 & 3 \\ 5 & 3 & 10 \\ \end{array} \right) $$

==============================================

$$\left( \begin{array}{rrr} 1 & - 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P = \left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & - 2 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q = \left( \begin{array}{rrr} 2 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D = \left( \begin{array}{rrr} 1 & 0 & 5 \\ 0 & - 4 & - 7 \\ 5 & - 7 & 10 \\ \end{array} \right) $$

==============================================

$$\left( \begin{array}{rrr} 1 & 0 & - 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P = \left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & - 2 & - 5 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q = \left( \begin{array}{rrr} 2 & 1 & 5 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 4 & - 7 \\ 0 & - 7 & - 15 \\ \end{array} \right) $$

==============================================

$$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & - \frac{ 7 }{ 4 } \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P = \left( \begin{array}{rrr} 0 & 1 & - \frac{ 7 }{ 4 } \\ 1 & - 2 & - \frac{ 3 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q = \left( \begin{array}{rrr} 2 & 1 & 5 \\ 1 & 0 & \frac{ 7 }{ 4 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 4 & 0 \\ 0 & 0 & - \frac{ 11 }{ 4 } \\ \end{array} \right) $$

==============================================

$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 0 & 1 & 0 \\ 1 & - 2 & 0 \\ - \frac{ 7 }{ 4 } & - \frac{ 3 }{ 2 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 0 & 2 & 3 \\ 2 & 1 & 5 \\ 3 & 5 & 10 \\ \end{array} \right) \left( \begin{array}{rrr} 0 & 1 & - \frac{ 7 }{ 4 } \\ 1 & - 2 & - \frac{ 3 }{ 2 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 4 & 0 \\ 0 & 0 & - \frac{ 11 }{ 4 } \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 2 & 1 & 0 \\ 1 & 0 & 0 \\ 5 & \frac{ 7 }{ 4 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 4 & 0 \\ 0 & 0 & - \frac{ 11 }{ 4 } \\ \end{array} \right) \left( \begin{array}{rrr} 2 & 1 & 5 \\ 1 & 0 & \frac{ 7 }{ 4 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 0 & 2 & 3 \\ 2 & 1 & 5 \\ 3 & 5 & 10 \\ \end{array} \right) $$

Will Jagy
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  • Very interesting properties you used and so is the link you provided about the Hermitian method for symmetric matrices – Anonymous196 Oct 26 '17 at 19:32
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Elementary matrices result from applying the elementary row operations to the identity matrix. Performing a row operation on a matrix is the same as multiplying on the left by the corresponding elementary matrix. The can play an important role in establishing many of the basic properties of matrices.

Bernard
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Rene Schipperus
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