UA-AUG-AUMAT229 Pre-read

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Section 9.1 Pre-read

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Subsection Definitions of some important matrix groups

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Definition 9.1.1. General linear group: ${GL}_{n} (R)$ .

The group of invertible $n \times n$ matrices with real entries, with multiplication as the group operation.

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Remark 9.1.2. For students who have taken AUMAT 220.

The linear part of the terminology stems from the fact that a general linear group consists of all invertible linear (matrix) operators on the vector space $R^{n} .$ An abstract vector space $V$ also has an associated general linear group $GL (V)$ of invertible linear operators, though this is a group of functions not of matrices. (Though as we learn in AUMAT 220, if $V$ is finite-dimensional and we have made a fixed choice of basis for $V,$ then to each linear operator on $V$ we can associate a matrix.)

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A

is an

n \times n

matrix and

x

is an

n \times 1

column vector in

R^{n},

then the result of computing

A x

will also be an

n \times 1

column vector in

R^{n} .

A

is invertible (i.e. if

A

is an element in

{GL}_{n} (R)

), then the mapping

x \mapsto A x

can be reversed, which means that “multiplication by $A$ ” defines a permutation of

R^{n} .

In other words, ${GL}_{n} (R)$ is a subgroup of $S_{R^{n}}$ .

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Remark 9.1.3.

There is no need to perform the Subgroup Test to verify that ${GL}_{n} (R)$ is a subgroup of $S_{R^{n}}$ — we know from AUMAT 120 that

the product of two invertible matrices is an invertible matrix, and
the inverse of an invertible matrix is an invertible matrix.

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In this chapter we will study several important subgroups of

{GL}_{n} (R) .

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Definition 9.1.4. Special linear group: ${SL}_{n} (R)$ .

The group consisting of those invertible $n \times n$ matrices that have determinant equal to $1 .$

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Remark 9.1.5.

The definition of this group via the determinant is an example of a more general pattern that we will encounter again in this course. From AUMAT 120 we know that the determinant of a product is the product of the determinants. This course has given us a more sophisticated way to think of this fact: the determinant is an Operation-preserving map map from ${GL}_{n} (R)$ to the multiplicative group $R^{\times}$ of nonzero real numbers. (Remember that a matrix is invertible precisely when its determinant is nonzero.) And ${SL}_{n} (R)$ is defined to consist of precisely those matrices in ${GL}_{n} (R)$ that this operation-preserving map sends to the identity element in $R^{\times} .$

Moreover, this pattern for operation-preserving maps on groups is one instance of a more widespread phenomenon in abstract algebra. Another instance of the same phenomenon is the fact that the kernel of a linear transformation between vector spaces is always subspace of the domain space, as we will learn (or have learned, as the case may be) in AUMAT 220.

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Definition 9.1.6. Orthogonal group: $O_{n} (R)$ .

The group consisting of those invertible $n \times n$ matrices that satisfy the condition

A^{T} A = I

(where $I$ is the $n \times n$ identity matrix, as usual).

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Here are some facts about orthogonal matrices from AUMAT 220. We will not discuss their proofs in this course; it would not be too difficult to verify these on your own, or you can just take them on faith.

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Fact 9.1.7.

A square matrix is orthogonal precisely when its columns form an orthonormal set. That is, as vectors in $R^{n},$ each column has length one and pairs of columns are orthogonal (have dot product equal to $0$ ).

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Fact 9.1.8.

Suppose $A$ is an orthogonal matrix. Then multiplication by $A$ against column vectors in $R^{n}$ preserves

dot product (that is, the value of the dot product between the vectors $A x$ and $A y$ is always equal to the value of the dot product between $x$ and $y$ );
length (that is, the length of the vector $A x$ is always equal to the value of the length of $x$ );
angle (that is, the angle between the vectors $A x$ and $A y$ is always equal to the angle between $x$ and $y$ ); and
orthogonality (that is, vectors $A x$ and $A y$ are orthogonal whenever $x$ and $y$ are orthogonal);

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Remark 9.1.9.

Thinking of $O_{n} (R)$ as a subgroup of $S_{R},$ Fact 9.1.8 severely restricts how elements of $O_{n} (R)$ can permute the vectors of $R^{n},$ as they must preserve all aspects of the geometric relationships between vectors.

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Definition 9.1.10. Special orthogonal group: ${SO}_{n} (R)$ .

The group consisting of those orthogonal $n \times n$ matrices that have determinant equal to $1 .$

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Note 9.1.11.

By definition, ${SO}_{n} (R)$ is a subgroup of both ${SL}_{n} (R)$ and $O_{n} (R)$ (and, of course, ${GL}_{n} (R)$ ). In fact, ${SO}_{n} (R)$ is the intersection of ${SL}_{n} (R)$ and $O_{n} (R),$ and we found in Discovery 5.9 that the intersection of two subgroups is again a subgroup.

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Subsection Geometry in $R^{2}$

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Recall that

\begin{aligned} e_{1} & = (1, 0), & e_{2} & = (0, 1) \end{aligned}

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are called the standard basis vectors for

R^{2} .

Every vector in

R^{2}

is naturally a linear combination of these two vectors:

(x, y) = x e_{1} + y e_{2} .

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In AUMAT 120, as part of our exploration of eigenvectors and diagonalization, we discovered that the collected results of matrix-times-standard-basis-vector calculations tells us about the matrix:

\begin{aligned} A e_{1} & = first column of A, \\ A e_{2} & = second column of A . \end{aligned}

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So, for a

2 \times 2

matrix

A,

if we know the results of these two calculations, we know the matrix

A .

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Remark 9.1.12.

Again, this is a specific instance of a more abstract pattern: in AUMAT 220 we will learn (or have learned) that a linear transformation between vector spaces is completely determined by how it transforms a fixed choice of basis for the domain space.

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Our exploration of the orthogonal group will rely on the matrix-times-standard-basis-vector pattern described above. Now, orthogonal matrices also preserve all aspects of geometry of

R^{n}

(Fact 9.1.8), so we would should pay special attention to the geometric properties of the standard basis. In particular, the standard basis is an orthonormal set: each standard basis vector has length

1,

and pairs of standard basis vectors are orthogonal (i.e. perpendicular). What other vectors have these properties?

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The head of a vector in

R^{2}

of length

1

will land on the unit circle when its tail is placed at the origin. And the unit circle is best described by trigonometry: by definition, the point on the unit circle at angle

θ

from the positive

x

-axis (counter-clockwise) has coordinates

(\cos θ, \sin θ) .

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Here is a diagram giving the coordinates of four vectors around the unit circle at intervals of

π / 2,

so that consecutive pairs are orthogonal.

Four vectors at $\pi/2$-radian intervals around the unit circle. — 🔗
Figure 9.1.13. Four vectors at $π / 2$ -radian intervals around the unit circle.

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To convince you that the coordinates of other three points (besides

(\cos θ, \sin θ)

) are labelled correctly, use the trig identities for

\cos (θ + \frac{π}{2}), \sin (θ + \frac{π}{2}),

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and the fact that oppositely directed vectors are negatives of one another.