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Orthogonal Diagonalization

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Wize University Linear Algebra Textbook > Orthogonality

Orthogonal Diagonalization of Symmetric Matrices (COMING SOON)

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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]
More Orthogonal Diagonalization of Symmetric Matrices (COMING SOON) Questions:
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
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] .