Sean Fitzpatrick |
---|
University of Lethbridge |
![]() |
Determine the matrix transformation that:
Reflects across the \(y\) axis, stretches vertically by a factor of 3, and then rotates by \(45^\circ\text{.}\)
Reflects across the line \(y=x\text{,}\) stretches vertically be a factor of 3, then reflects across the \(y\) axis.
The transpose of a matrix swaps its rows and columns.
If \(A=[a_{ij}]\) is an \(m\times n\) matrix, then \(A^T\) is the \(n\times m\) matrix whose \((i,j)\) entry is \(a_{ji}\text{.}\)
Examples: on the board.
Why should we care about transpose?
Admittedly, it's one of several things we teach you to compute in Math 1410, and make you wait until later courses to really understand.
Later, it gets related to dual vectors and dual operators, and is important in things like quantum mechanics.
It gives us an easy way to turn columns into rows and vice versa: sometimes we write \(\bbm 1\amp 2\amp 3\ebm^T\) instead of \(\bbm 1\\2\\3\ebm\) because it fits better on the page.
It also connects with the dot product: we can write \(\vec{v}\cdot\vec{w} = (\vec{v})^T\vec{w}\text{.}\)
In each case, assume that \(A\) and \(B\) have the right size for the operation to be defined.
(True if both are \(n\times n\) but things work for non-square matrices too.)
\((A+B)^T = A^T+B^T\)
\((kA)^T = kA^T\)
\((AB)^T=B^TA^T\)
\((A^T)^T=A\)
\((A^T)^{-1}=(A^{-1})^T\text{,}\) if \(A\) is invertible.
An \(n\times n\) matrix \(A\) is symmetric if
What can we say about the entries of \(A\) in each case?
Show that \(A+A^T\) is symmetric, and \(A-A^T\) is antisymmetric, for any \(n\times n\) matrix \(A\)
Show that \(AA^T\) and \(A^TA\) are symmetric for any matrix \(A\text{.}\)
Show that any square matrix \(A\) can be written as the sum of a symmetric and an antisymmetric matrix.
The trace of a matrix is simply the sum of its diagonal entries.
If \(A = [a_{ij}]\) is an \(n\times n\) matrix, then
(If \(A\) is \(m\times n\) with \(m\neq n\text{,}\) sum to whichever of \(m,n\) is smaller.)
Examples? On the board!
\(\tr(A+B) = \tr(A)+\tr(B)\)
\(\tr(kA) = k\tr(A)\)
\(\tr(AB)=\tr(BA)\) (as long as both products are defined)
\(\tr(A^T)=\tr(A)\)
Note: on the set of all \(m\times n\) matrices, the pairing
The determinant is a function that assigns a number to any square matrix \(A\text{.}\)
We denote this number by \(\det A\) or \(\abs{A}\text{.}\)
We'll define \(\det A\) recursively, starting with \(2\times 2\) matrices, and then showing how to reduce the determinant of a larger matrix to smaller ones.
There is a general formula, but it's........ complicated.
You've already learned how to compute these, back when we did cross products!
Given \(A=\bbm a\amp c\\b\amp d\ebm\text{,}\) \(\det A = ad-bc\text{:}\)
Note:
If \(\vec{v}=\bbm a\\b\ebm\) is parallel to \(\vec{v}\bbm c\\d\ebm\text{,}\) then \(\det A = 0\text{.}\)
Otherwise, \(\det A\) calculates (up to sign) the area of the parallelogram spanned by \(\vec v\) and \(\vec w\text{.}\)
This is also not a big leap from cross products:
You might recall that this is the same as the scalar triple product \(\vec{a}\cdot (\vec{b}\times \vec{c})\text{.}\)
Example: compute the determinant of \(A = \bbm 2\amp -1\amp 3\\0\amp 4\amp 1\\-1\amp 0\amp 5\ebm\text{.}\)
Given an \(n\times n\) matrix \(A = [a_{ij}]\text{,}\)
The \((i,j)\) minor of \(A\) is the \((n-1)\times (n-1)\) matrix \(M_{ij}\) obtained by deleting row \(i\) and column \(j\) of \(A\)
The \((i,j)\) cofactor of \(A\) is the number \(C_{ij}\) defined by
Note: \((-1)^{i+j}\) equals \(+1\) if \(i+j\) is even, and \(-1\) if \(i+j\) is odd.
(Now do some examples, Sean)
In general, we define \(\det A = a_{11}C_{11}+a_{12}C_{12}+\cdots + a_{1n}C_{1n}\) (cofactor expansion along first row). (Each cofactor \(C_{1j}\) is a determiant one smaller than \(A\))
Then we prove (OK, state assertively) that we can actually expand any row or column of \(A\text{.}\)
Then we'll observe that determinants of triangular matrices are really easy.
Finally, we'll see what happens to the determinant if we use row operations to get a matrix into triangular form.