Section 6.3

Orthogonal Matrices

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

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Orthogonal Matrices

An nxn matrix is said to be an orthogonal matrix if its columns form an orthonormal set.

Example 1: Consider the orthonormal basis of

> u1:=vector([3/5,4/5,0]);
u2:=vector([-4/5,3/5,0]);
u3:=vector([0,0,1]);

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Construct the matrix whose columns are the vectors

> A:=augment(u1,u2,u3);

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The columns of this matrix form an orthonormal set. Thus A is called an

Orthogonal Matrix

Let us study some of its properties :

Take for example the transpose of A

> trA:=transpose(A);

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What is the product of A and ?

> AtrA=multiply(A,trA);

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What is the determinant of the matrix A?

What is the inverse of the orthogonal matrix A?

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In general, an orthogonal matrix has the following properties:

1. Its inverse is equal to its transpose, i.e. and

2. Its determinant is equal to +1 or -1.

3. The rows of form an orthonormal basis for

4. The columns of form an orthonormal basis for

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Example 2: Preservation of dot products, norms and angles.

> x:=vector([1, -1, 2]);
y:=vector([-2, 1, 3]);

Dot product of x and y:

> innerprod(x,y);

Dot product of Ax and Ay:

> innerprod(multiply(A,x),multiply(A,y));

Norm of x and Ax:

> `||x||`=sqrt(innerprod(x,x));

> `||Ax||`=sqrt(innerprod(multiply(A,x),multiply(A,x)));

Angle between x and y is the same as the angle between Ax and Ay.

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Eigenvectors of a real symmetric matrix that correspond to different eigenvlaues are orthogonal.

Moreover, every symmetric matrix is diagonalizable, where can be chosen to

be an orthogonal matrix. Hence, .

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Example 3: Consider the matrix

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> A:=matrix([[0,1,1],[1,2,1],[1,1,0]]);

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> eigenvals(A);

> eigenvects(A);

Eigenvectors are given as:

> v1 := vector([1,-1,1]);
v2 := vector([1,2,1]);
v3 := vector([-1,0,1]);

Construct matrix C whose columns are the vectors

> C := augment(v1,v2,v3);

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Product with C

> multiply(transpose(C),C);

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We want this product to be equal to I. Want is wrong? The columns do not

have length 1.

> C := augment(v1/sqrt(3),v2/sqrt(6),v3/sqrt(2));

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> multiply(transpose(C),C);

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Product

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> multiply(transpose(C),A,C);

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Exercises

1-17 odd.