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Wize University Linear Algebra Textbook > Orthogonality

Orthogonal Diagonalization of Symmetric Matrices (COMING SOON)

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Extra Practice
Example: Orthogonal Diagonalization
Orthogonally diagonalize the matrix A=[−5−22−2−2−42−4−2]A=\left[\begin{array}{rrr}-5&-2&2\\-2&-2&-4\\2&-4&-2\end{array}\right]
Orthogonal Matrices
Prove that multiplying a vector by an orthogonal matrix preserves its length, i.e. for PP orthogonal, ∣∣Px⃗∣∣=∣∣x⃗∣∣||P\vec x||=||\vec x|| for all vectors x⃗\vec x.
Orthogonal Diagonalization
For each of the two following matrices, determine if they are orthogonally diagonalizable, and if so, find the corresponding diagonal matrix DD and the orthogonal diagonalizing matrix PP .
a) A:=[−11−111−11−1−11−111−11−1]A:=\left[\begin{array}{rrrr}-1&1&-1&1\\1&-1&1&-1\\-1&1&-1&1\\1&-1&1&-1\end{array}\right]
b) B:=[11−11−1111111−11−111]B:=\left[\begin{array}{rrrr}1&1&-1&1\\-1&1&1&1\\1&1&1&-1\\1&-1&1&1\end{array}\right]
Orthogonal Diagonalization
Orthogonally diagonalize the matrix A:=[01101001100−101−10]A:=\left[\begin{array}{rrrr}0&1&1&0\\1&0&0&1\\1&0&0&-1\\0&1&-1&0\end{array}\right] , or prove that it is not possible to do so.
Practice Question: Orthogonal Diagonalization
Orthogonally diagonalize the matrix A=[10−2010−201]A=\left[\begin{array}{rrr}1&0&-2\\0&1&0\\-2&0&1\end{array}\right] .