For each of the matrices A in Exercises 7 through 11, find...

For each of the matrices $A$ in Exercises 7 through $11,$ find an orthogonal matrix S and a diagonal matrix $D$ such that $S^{-1} A S=D .$ Do not use technology.
$$A=\left[\begin{array}{rrr}1 & -2 & 2 \\-2 & 4 & -4 \\2 & -4 & 4\end{array}\right]$$

Key Concepts

Matrix Diagonalization

Matrix diagonalization is the process of finding an invertible matrix S and a diagonal matrix D such that S?¹AS = D. This process simplifies many computations by transforming the matrix into a form where its action is easy to understand, with the diagonal entries of D being the eigenvalues of A.

Orthogonal Diagonalization

Orthogonal diagonalization is a specialized form of diagonalization applicable to symmetric matrices. It constructs an orthogonal matrix S (where S?¹ = S?) such that S?AS = D. This method takes advantage of the fact that symmetric matrices have orthonormal eigenvectors, making computations more stable and preserving lengths and angles.

Eigenvalues

Eigenvalues are the scalars ? that satisfy the equation det(A - ?I) = 0 for a matrix A. They represent the factors by which the corresponding eigenvectors are scaled under the transformation represented by A, and they are essential for determining the diagonal entries of the diagonal matrix during the process of diagonalization.

Eigenvectors

Eigenvectors are the nonzero vectors v that satisfy the equation A*v = ?*v for a given eigenvalue ?. In the process of diagonalizing a matrix, especially when forming an orthogonal basis, the eigenvectors serve as the columns of the orthogonal matrix S, and they reveal the invariant directions of the transformation.

Spectral Theorem

The spectral theorem states that every real symmetric matrix can be diagonalized by an orthogonal matrix. This theorem is foundational in understanding how matrices can be represented in a basis where they act by simple scalar multiplication, and it underpins techniques used in many areas of mathematics and physics.

Transcript

00:01 All right, so here we have our matrix a is our matrix 1, negative 2, negative 2, 4, 4, 4.

00:15 And we want to find an orthogonal matrix s and a diagonal matrix d, such that we have s inverse times as is equal to d.

00:28 So we can solve the equation here.

00:30 We take the determinant of a minus lambda i, where i is our identity, set equal to zero.

00:36 So what we're going to have is one minus lambda, two, two, and then negative two, four minus lambda, negative four, and then two, negative four, and then four minus lambda.

00:54 Taking the determinant and setting it equal to zero.

00:57 All right, and this is going to end up giving us a negative lambda cube plus 9 lambda squared, which we can factor as negative lambda squared times lambda minus 9 equals 0.

01:14 So we can see that the eigenvalues here, we have lambda equals 0, has multiplicity 2, and lambda equals 9 is going to have multiplicity 1.

01:25 Okay, so then we take, for each eigenvalue, you find the corresponding eigenspice.

01:31 So for the lambda equals zero, we find the corresponding eigen space by just reducing the matrix a minus lambda i to row reduced row rest long form.

01:42 So we start with, so 1 minus 0, we get 1, negative 2, 2, 2, 4, negative 4, 2, 2, 4, 4, and then we reduce that to 1 .92.

01:55 Negative 2, 2 with all zeros.

01:59 Next two rows.

02:04 And therefore, then the eigenspace for e1, for lambda equals 0, we take the kernel here and to the kernel of v1, v2, v3...

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