For each of the following symmetric matrices A, find a...

For each of the following symmetric matrices $A,$ find a nonsingular matrix $P$ such that $D=P^{T} A P$ is diagonal: (a) $A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 3 & 6 \\ 2 & 6 & 7\end{array}\right]$ (b) $A=\left[\begin{array}{rrr}1 & -2 & 1 \\ -2 & 5 & 3 \\ 1 & 3 & -2\end{array}\right]$ (c) $A=\left[\begin{array}{rrrr}1 & -1 & 0 & 2 \\ -1 & 2 & 1 & 0 \\ 0 & 1 & 1 & 2 \\ 2 & 0 & 2 & -1\end{array}\right]$

For each of the following symmetric matrices $A,$ find a nonsingular matrix $P$ such that $D=P^{T} A P$ is diagonal:
(a) $A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 3 & 6 \\ 2 & 6 & 7\end{array}\right]$
(b) $A=\left[\begin{array}{rrr}1 & -2 & 1 \\ -2 & 5 & 3 \\ 1 & 3 & -2\end{array}\right]$
(c) $A=\left[\begin{array}{rrrr}1 & -1 & 0 & 2 \\ -1 & 2 & 1 & 0 \\ 0 & 1 & 1 & 2 \\ 2 & 0 & 2 & -1\end{array}\right]$

Key Concepts

Symmetric Matrices

A symmetric matrix is one which equals its transpose, meaning its entry in the i-th row and j-th column is equal to that in the j-th row and i-th column. This property guarantees that all eigenvalues are real and underpins many powerful results in linear algebra, making symmetric matrices particularly amenable to diagonalization and other forms of analysis.

Spectral Theorem

The spectral theorem states that every real symmetric matrix can be diagonalized using an orthogonal transformation, meaning there exists an orthogonal matrix whose columns are the eigenvectors of the original matrix, and that transforms the matrix into a diagonal form. This theorem is fundamental in understanding the structure of symmetric matrices and in simplifying quadratic forms.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors provide a way to understand a linear transformation by identifying invariant directions and scaling factors. For symmetric matrices, the eigenvalues are guaranteed to be real and the eigenvectors can be chosen to be orthogonal, which is crucial in forming the change-of-basis that diagonalizes the matrix.

Congruence Transformation

A congruence transformation involves finding a nonsingular matrix P such that P^T A P is diagonal. In the case of symmetric matrices, this approach is closely related to the orthogonal diagonalization provided by the spectral theorem. It is particularly useful in simplifying quadratic forms and studying the matrix’s inertia, and while P can be chosen orthogonally, the general requirement is simply nonsingularity.

Transcript

00:06 For 34a, we have m equal to ai, giving us 1 .02, 036, 26, 7, in our identity matrix.

00:25 Now we're going to have row 3 minus 2 of row 1 to 0 out row 1.

00:30 Then we're going to have column 3 minus 2, column 1.

00:38 So that's going to produce 1 -0 -0 -0 0 -0.

00:46 We have 3 and 6 here.

00:48 R row 3 minus 2 row 1.

00:51 We have 6 minus 0, and we hit 7 minus 4, which is 3.

01:04 Then we have 1 -1, but this negative 2 from earlier is there now.

01:14 And next we're going to have the following row 3 minus 2 row 2 and we're going to have column 3 minus 2 column 2 so now we'll have 1 0 0 0 0 0 0 0 and this will be 1 0 0 0 0 0 0 0 0 and 1.

02:04 So now we have our diagonal and we have p transpose.

02:11 So we can write this as p equals, remember this is d, and this is p transpose, is 1, 0, 0 ,0, 0, 0, 0, 0, 0, 0, 0, and 0, 1.

02:27 And d is going to equal our diagonal of 1, 3, and negative 9.

02:38 Now for b, m equals ai again, so we have 1, negative 2, 1 ,000, 1 ,000.

02:54 015 and 05 negative 3 multiplied by the identity matrix.

03:06 And so we're going to have 2 row 1.

03:09 Oh, i'm sorry.

03:11 I miswrote this.

03:24 It's going to be negative 2, 5, 3, 1, 3, 13, negative 2.

03:30 Two, we're going to have two row one plus row two and row three minus row one to zero at that column.

03:42 We're also going to do two column one plus column two and column three minus column one.

03:58 So if we look at row one and row two, that's going to change this to 015.