Tag Archives: determinant

SL(2,IR) is the Commutator Subgroup of GL(2,IR)

Here is a proof of the above fact.

Let $N$ be the commutator subgroup of the general linear group $GL(2,\mathbb R)$; i.e.,

$N=\langle ABA^{-1}B^{-1}:A,B\in GL(2,\mathbb R)\rangle$.

First, it is clear that $N$ is contained in the special linear group $SL(2,\mathbb R)$, since $\det(ABA^{-1}B^{-1})=1$ for any $A,B\in GL(2,\mathbb R)$. Next, we claim that $N$ contains all matrices

$\begin{pmatrix} 1 & b\\ 0 & 1\end{pmatrix}$.

This follows from noting that

$\begin{pmatrix} 1 & b\\ 0 & 1\end{pmatrix}=\begin{pmatrix} 1 & b\\ 0 & b\end{pmatrix}\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1 & b\\ 0 & b\end{pmatrix}^{-1}\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}^{-1}$.

By taking transposes, it also follows that $N$ contains all matrices

$\begin{pmatrix} 1 & 0\\ c & 1\end{pmatrix}$.

Further, $N$ contains all matrices

$\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}$

since

$\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}=\begin{pmatrix} a & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}\begin{pmatrix} a & 0\\ 0 & 1\end{pmatrix}^{-1}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}^{-1}$

for any $a\neq 0$.

Now let

$\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in SL(2,\mathbb R)$.

Then $ad-bc=1$. Using the above results,

$\begin{pmatrix} a & b\\ c & d\end{pmatrix}=\begin{pmatrix} 1 & 0\\ c/a & 1\end{pmatrix}\begin{pmatrix} 1 & ab\\ 0 & 1\end{pmatrix}\begin{pmatrix} a & 0\\ 0 & 1/a\end{pmatrix}\in N$

if $a\neq 0$, and

$\begin{pmatrix} a & b\\ c & d\end{pmatrix}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}1&-d/b\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ ab&1\end{pmatrix}\begin{pmatrix}1/b&0\\ 0&b\end{pmatrix}\in N$

if $b\neq 0$, and the latter since

\begin{aligned}&\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}\\=&\begin{pmatrix}x&y\\0&-x-y\end{pmatrix}\begin{pmatrix}-x-y&0\\ x&y\end{pmatrix}\begin{pmatrix}x&y\\0&-x-y\end{pmatrix}^{-1}\begin{pmatrix}-x-y&0\\ x&y\end{pmatrix}^{-1}\\ \in &\ N\end{aligned}

for any $x,y,x+y\neq 0$. Thus $SL(2,\mathbb R)\subseteq N$, i.e., $N=SL(2,\mathbb R)$.

Filed under Linear Algebra

Some Interesting Linear Algebra Proofs

Below are some cute linear algebra results and proofs cherrypicked from various sources. All the standard hypotheses (on the base field, the size of the matrices, etc.) that make the claims valid are assumed. The list will likely be updated.

Fact 1. Let $A,B,X$ be matrices. If $AX=XB$, then $p(A)X=Xp(B)$ for any polynomial $p$.

Proof. We have $A^2X=A(AX)=A(XB)=(AX)B=(XB)B=XB^2$. By induction, $A^nX=XB^n$ for any $n\in\mathbb N$. Hence the result follows. $\square$

Remark. Note that $X$ need not be square, let alone invertible.

Fact 2. Let $x_0,\dots,x_n$ be distinct. Then the Vandermonde matrix

$\displaystyle V=\begin{pmatrix} 1 & x_0 & \cdots & x_0^n\\ 1 & x_1 & \cdots & x_1^n\\ \vdots & \vdots & \ddots & \vdots\\ 1 & x_n & \cdots & x_n^n\end{pmatrix}$

is invertible.

Proof. It suffices to show that the kernel of the linear transformation $f(x)=Vx$ is trivial. If $a=(a_0,\dots,a_n)$ is in the kernel, then $p(X)=a_0+a_1X+\cdots+a_nX^n=0$ for each $X\in\{x_0,\dots,x_n\}$. Since $\deg(p)=n$, this forces $p(X)$ to be identically zero. Thus $a=0$. $\square$

Fact 3. A matrix is diagonalisable iff its minimal polynomial decomposes into distinct linear factors.

Proof. A matrix is diagonalisable iff every Jordan block has size $1$. Since the multiplicity of an eigenvalue in the minimal polynomial corresponds to the size of the largest Jordan block, the result follows. $\square$

Corollary. Idempotent matrices are diagonalisable. Moreover, the rank of an idempotent matrix is equal to the algebraic multiplicity of the eigenvalue $1$.

Fact 4. If $A^nx=0$ and $A^{n-1}x\neq 0$, then $x,Ax,\dots,A^{n-1}x$ are linearly independent.

Proof. Note that $\ker(A^n)\supseteq\ker(A^{n-1})$. Let $x\in\ker(A^n)\setminus\ker(A^{n-1})$ and suppose that $a_0x+a_1Ax+\cdots+a_{n-1}A^{n-1}x=0$. Multiplying both sides by $A^i$ for $i=n-1,\dots,1$ shows that $a_i=0$ for all $i$, as desired. $\square$

Corollary 1. If $\ker(A^n)\neq\ker(A^{n-1})$, then $\dim(\ker(A^j))\ge j$ for $j=1,\dots,n$.

Proof. If $x\in\ker(A^n)\setminus\ker(A^{n-1})$, then $A^{n-1}x,\dots,A^{n-j}x\in\ker(A^j)$. $\square$

Corollary 2. If $A$ is $n\times n$, and $\ker(A^n)\neq\ker(A^{n-1})$, then $\dim(\ker(A^j))=j$ for each $0\le j\le n$. In particular, $A$ is similar to the nilpotent Jordan block of size $n$.

Fact 5. If $f$ is linear on $V$, then $V\cong\ker(f)\oplus f(V)$.

Proof. The short exact sequence

$0\to\ker(f)\hookrightarrow V\twoheadrightarrow f(V)\to 0$

is split. So the result follows by the splitting lemma. $\square$

Corollary. If $W\subseteq V$ is a subspace, then $V\cong W\oplus W^\perp$.

Fact 6. If $A$ is $n\times n$, then $r(A)\ge n-k$, where $k$ is the algebraic multiplicity of the eigenvalue $0$ of $A$.

Proof. Since the nullity of $A$ is the geometric multiplicity of the eigenvalue $0$, and the geometric multiplicity of an eigenvalue is at most its algebraic multiplicity, we get $r(A)=n-n(A)\ge n-k$. $\square$

Fact 7. The number of distinct eigenvalues of $A$ is at most $r(A)+1$.

Proof. The rank of a matrix is the number of non-zero eigenvalues and the nullity is the number of zero eigenvalues, both counted with multiplicity. $\square$

Filed under Linear Algebra

Discriminants and Lattices

Let $K=\mathbb Q(\alpha)$ be a quadratic number field. For $a,b\in K$, recall that the discriminant $\Delta(a,b)$ is defined as

$\displaystyle \Delta(a,b):=\left|\begin{matrix} a^{(1)} & a^{(2)}\\ b^{(1)} & b^{(2)}\end{matrix}\right|^2$

where $a^{(1)}, a^{(2)}$ are the Galois conjugates of $a$ and $b^{(1)}, b^{(2)}$ are those of $b$. For any $\beta\in K$ we define its discriminant to be $\Delta(\beta):=\Delta(1,\beta)$.

Write $a=a_1+a_2\alpha$ and $b=b_1+b_2\alpha$. Then

$\left(\begin{matrix} a\\ b\end{matrix}\right)=\underbrace{\left(\begin{matrix} a_1 & a_2\\ b_1 & b_2\end{matrix}\right)}_{A}\left(\begin{matrix} 1\\\alpha\end{matrix}\right)$

If $\alpha,\bar\alpha$ are the Galois conjugates of $\alpha$, then

$\Delta(a,b)=\left|\begin{matrix} a_1+a_2\alpha & a_1+a_2\bar\alpha\\ b_1+b_2\alpha & b_1+b_2\bar\alpha\end{matrix}\right|^2=\left|\begin{matrix} a_1 & a_2\\ b_1 & b_2\end{matrix}\right|^2\left|\begin{matrix} 1 & 1\\\alpha & \bar\alpha\end{matrix}\right|^2$

$\therefore\boxed{\Delta(a,b)=(\det A)^2\Delta(\alpha)}$

Now suppose that $\mathbb Z[\alpha]=a\mathbb Z+b\mathbb Z$. Then $\mathbb Z[\alpha]$ is spanned by $\{a,b\}$, so there are integers $p,q,r,s$ such that

$\underbrace{\left(\begin{matrix} p & q\\ r & s\end{matrix}\right)}_{M}\left(\begin{matrix} a\\ b\end{matrix}\right)=\left(\begin{matrix} 1\\\alpha\end{matrix}\right)$

So we have

$MA\left(\begin{matrix} 1\\\alpha\end{matrix}\right)=\left(\begin{matrix} 1\\\alpha\end{matrix}\right)$.

Lemma. If $P$ is a $2\times 2$ matrix with integer coefficients and $w=(1,\alpha)^T$ with $\alpha\not\in\mathbb Q$, then $Pw=w$ if and only if $P=I$, the $2\times 2$ identity matrix.

Proof. This follows from the $\mathbb Z$-linear independence of $\{1,\alpha\}$. More concretely,

$\underbrace{\left(\begin{matrix} s & t\\ u & v\end{matrix}\right)}_{P}\left(\begin{matrix}1\\\alpha\end{matrix}\right)=\left(\begin{matrix} 1\\\alpha\end{matrix}\right)\Rightarrow\begin{cases}s+t\alpha=1\\ u+v\alpha=\alpha\end{cases}$

$\therefore s=1,\ t=0,\ u=0,\ v=1\Rightarrow P=I$. $\square$

Thus $MA=I$, so that $\det(M)\det(A)=1$. But $\det(M)$ and $\det(A)$ are integers. Hence $|\det(M)|=|\det(A)|=1$, i.e. $\Delta(a,b)=\Delta(\alpha)$. Thus

Fact. $\{a,b\}\subset\mathbb Z[\alpha]$ spans $\mathbb Z[\alpha]$ if and only if $\Delta(a,b)=\Delta(\alpha)$.

Note that all of the above arguments generalize to arbitrary number fields.

A nice corollary:

Corollary. $(a,b)$ and $(c,d)$ generate $\mathbb Z^2$ (as a group) if and only if

$\left|\begin{matrix} a & b\\ c & d\end{matrix}\right|=\pm 1$.

In other words, two bases generate the same lattice only if their fundamental parallelograms have equal areas.

1 Comment

Filed under Geometry, Linear Algebra, Number Theory