Calculus: Find the Eigenvalues, Eigenvectors, and Orthogonal Matrix that Diagonalizes the 2x2 Symmetric Matrix

Find the eigenvalues, eigenvectors, and orthogonal matrix that diagonalizes the 2×2 symmetric matrix A

A =

1	2
2	4

Because A is a 2×2 symmetric matrix, we can find the eigenvalues using the following formulas:

μ+ = (a+d)/2 + sqrt(b^2 + ((a-d)/2)<>2) and μ- = (a+d)/2 – sqrt(b2 + ((a-d)/2)<>2)
μ+ = (1+4)/2 + sqrt(22 + ((1-4)/2)<>2) and μ- = (1+4)/2 – sqrt(22 + ((1-4)/2)^2)
μ+ = 5/2 + 5/2 and μ- = 5/2 – 5/2
μ+ = 5 and μ- = 0

Now that we have the eigenvalues, we can find the eigenvectors of A.  Because A is a 2×2 matrix, we can find these eigenvectors using the following formulas:

U1 = (1/|r1|) * vector orthogonal to r1
U2 = vector orthogonal to U1

where

A – μ+I =

r1

r2

First we find r1 by subtracting μ+ times the identity matrix from A.

A – μ+I =

-4	2
2	-1

which means r1 = (-4, 2)

Now we find the first eigenvector (U1) by multiplying the vector orthogonal to r1 by 1 over the dot product of r1

1/|r1| = 1/sqrt((-4)<>2 + 22) -> 1/(2*sqrt(5))

The vector orthogonal to r1 can be found by flipping the vector values of r1 then multiplying the top one by -1.

Vector orthogonal to r1 = (-2, -4)

U1 = (1/|r1|) * vector orthogonal to r1 = (1/(2sqrt(5)) * (-2,-4) -> (1/sqrt(5))(-1,-2)
U2 = vector orthogonal to U1 = (1/sqrt(5))*(2,-1)

The orthogonal matrix that diagonalizes A can be found by:

U = {U1, U2}

U = (1/sqrt(5))*

-1	2
-2	-1

**Solution: Eigenvalues: μ+ = 5 and μ- = 0 Eigenvectors: U1 = (1/sqrt(5))*(-1,-2) and U2 = (1/sqrt(5))*(2,-1) Orthogonal matrix that diagonalizes A: U = (1/sqrt(5))***

-1	2
-2	-1