In Exercises 25 and 26, mark each statement True or False....

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer. a. An $n \times n$ matrix that is orthogonally diagonalizable must be symmetric. b. If $A^{T}=A$ and if vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $A \mathbf{u}=3 \mathbf{u}$ and $A \mathbf{v}=4 \mathbf{v},$ then $\mathbf{u} \cdot \mathbf{v}=0$ c. $\operatorname{An} n \times n$ symmetric matrix has $n$ distinct real eigenval- ues. d. For a nonzero $\mathbf{v}$ in $\mathbb{R}^{n},$ the matrix $\mathbf{v}^{T}$ is called a projection matrix.

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer.
a. An $n \times n$ matrix that is orthogonally diagonalizable must be symmetric.
b. If $A^{T}=A$ and if vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $A \mathbf{u}=3 \mathbf{u}$ and $A \mathbf{v}=4 \mathbf{v},$ then $\mathbf{u} \cdot \mathbf{v}=0$
c. $\operatorname{An} n \times n$ symmetric matrix has $n$ distinct real eigenval-
ues.
d. For a nonzero $\mathbf{v}$ in $\mathbb{R}^{n},$ the matrix $\mathbf{v}^{T}$ is called a projection matrix.

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues are scalars linked to a matrix A through the equation A? = ??, where ? is an eigenvector. These concepts are crucial in understanding matrix behaviors, such as diagonalization. In symmetric matrices, even though eigenvectors corresponding to distinct eigenvalues are orthogonal, having distinct eigenvalues is not a necessary condition for symmetry.

Projection Matrices

Projection matrices are defined as matrices that satisfy the condition P² = P, meaning they are idempotent. They project vectors onto a subspace and are typically constructed with careful normalization. A mere transpose of a vector does not fulfill the properties required to be considered a projection matrix.

Orthogonal Diagonalization

Orthogonal diagonalization is the process of decomposing a matrix A into the form QDQ?, where Q is an orthogonal matrix and D is a diagonal matrix. This concept is central because it is equivalent to a matrix being symmetric, ensuring real eigenvalues and an orthonormal set of eigenvectors.

Symmetric Matrices

Symmetric matrices are those that satisfy A = A?. This property guarantees that all eigenvalues are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Such characteristics are essential for many theoretical and practical results in linear algebra, including diagonalizability and stability.

Transcript

00:01 We are given statements and we are asked to label them true or false and to explain.

00:09 In part a, the statement is that an n by n matrix that is orthogonally diagonalizable must be symmetric.

00:21 This statement is true.

00:28 This follows from theorem 2 from this chapter, which says that an n by n matrix a is orthogonally diagonalizable if and only if.

00:44 A is a symmetric matrix.

00:52 In part b, the statement is, if a transpose equals a, and if vectors u and v satisfy au equals 3u and av equals 4v, then u dotted with v equals 0.

01:12 This statement is true.

01:19 Notice that since a transpose equals a, we have that a is symmetric, and therefore, by theorem 1, from this chapter, it follows that eigenvectors from distinct eigenspaces are orthogonal to each other, and we have that since a .u is equal to 3u, and av is equal to 4v, we have that u and v and v are eigenvectors of a, associated with the eigenvectors or eigenvalues, 3 .3 .3.

02:49 And 4 respectively.

03:01 These are distinct eigenvalues, so by theorem 1, it follows that u1 is orthogonal to, or u is orthogonal to v, and so it follows that the dot product of u and v is going to be 0.

03:27 In part c, the statement is an n -by -n symmetric matrix has n distinct real eigenvalues.

03:35 This is false.

03:47 Indeed, an example in this section included a symmetric matrix a, which did not have n distinct real eigenvalues...

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