High School
SAT
ACT
University
MCAT
LSAT
DAT

Orthogonal Matrices

Related Topics

Wize University Linear Algebra Textbook > Orthogonality

Orthogonal Diagonalization of Symmetric Matrices (COMING SOON)

0 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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.
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 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] .