1. ENGG2013 Unit 19 The principal axes theorem
Mar, 2011.

2. Outline Special matrices Principle axes theorem
Symmetric, skew-symmetric, orthogonal
Principle axes theorem
Application to conic sections
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ENGG2013

3. Diagonalizable ??
A square matrix M is called diagonalizable if we can find an invertible matrix, say P, such that the product P–1 M P is a diagonal matrix.
Example
Some matrix cannot be diagonalized.
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ENGG2013

4. Theorem
An nn matrix M is diagonalizable if and only if we can find n linear independent eigenvectors of M.
Proof: For concreteness, let’s just consider the 33 case.
The three columns are linearly independent because the matrix is invertible
\begin{bmatrix}
x_1 & x_2 & x_3\\
y_1 & y_2 & y_3\\
z_1 & z_2 & z_3\\
\end{bmatrix}^{-1}\begin{bmatrix}
a & b & c\\
d & e & f\\
g & h & i
\end{bmatrix} \begin{bmatrix}
\end{bmatrix}=\begin{bmatrix}
\lambda_1 & & \\
& \lambda_2& \\
& & \lambda_3
\end{bmatrix}
by definition
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ENGG2013

5. Proof continued and and \begin{bmatrix} a & b & c\\ d & e & f\\
g & h & i
\end{bmatrix} \begin{bmatrix}
x_1 & x_2 & x_3\\
y_1 & y_2 & y_3\\
z_1 & z_2 & z_3\\
\end{bmatrix}=\begin{bmatrix}
\end{bmatrix}\begin{bmatrix}
\lambda_1 & & \\
& \lambda_2& \\
& & \lambda_3
\end{bmatrix}
kshum
ENGG2013

6. Complex eigenvalue
There are some matrices whose eigenvalues are complex numbers.
For example: the matrix which represents rotation by 45 degree counter-clockwise.
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ENGG2013

7. Theorem
If an nn matrix M has n distinct eigenvalues, then M is diagonalizable
\begin{bmatrix}
a & b & c\\
d & e & f\\
g & h & i
\end{bmatrix}\begin{bmatrix}
x_k\\
y_k\\
z_k\\
\end{bmatrix}= \lambda_k \begin{bmatrix}
\end{bmatrix} \text{ for }k=1,2,3
The converse is false: There is some diagonalizable matrix with repeated eigenvalues.
kshum
ENGG2013

8. Matrix in special form Symmetric: AT=A. Skew-symmetric: AT= –A.
Orthogonal: AT =A-1, or equivalently AT A = I.
Examples:
symmetric
and
orthogonal
symmetric
skew-symmetric
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ENGG2013

9. Orthogonal matrix A matrix M is called orthogonal if
Each column has norm 1 MT I M
Dot product = 1
\begin{bmatrix}
m_{11} & m_{12} & m_{13} & \cdots & m_{1n}\\
m_{21} & m_{22} & m_{23} & \cdots & m_{2n}\\\
m_{31} & m_{32} & m_{33} & \cdots & m_{3n}\\
\vdots & \vdots & \vdots & \ddots & \\
m_{n1} & m_{n2} & m_{n3} & \cdots & m_{nn}
\end{bmatrix}\begin{bmatrix}
m_{11} & m_{21} & m_{31} & \cdots & m_{n1}\\
m_{12} & m_{22} & m_{32} & \cdots & m_{n2}\\\
m_{13} & m_{23} & m_{33} & \cdots & m_{n3}\\
m_{1n} & m_{2n} & m_{3n} & \cdots & m_{nn}
\end{bmatrix}
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10. Orthogonal matrix A matrix M is called orthogonal if
Any two distinct columns are orthogonal
Dot product = 0
\begin{bmatrix}
m_{11} & m_{12} & m_{13} & \cdots & m_{1n}\\
m_{21} & m_{22} & m_{23} & \cdots & m_{2n}\\\
m_{31} & m_{32} & m_{33} & \cdots & m_{3n}\\
\vdots & \vdots & \vdots & \ddots & \\
m_{n1} & m_{n2} & m_{n3} & \cdots & m_{nn}
\end{bmatrix}\begin{bmatrix}
m_{11} & m_{21} & m_{31} & \cdots & m_{n1}\\
m_{12} & m_{22} & m_{32} & \cdots & m_{n2}\\\
m_{13} & m_{23} & m_{33} & \cdots & m_{n3}\\
m_{1n} & m_{2n} & m_{3n} & \cdots & m_{nn}
\end{bmatrix}
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11. Principal axes theorem
Given any nn symmetric matrix A, we have:
The eigenvalues of A are real.
A is diagonalizable.
We can pick n mutually perpendicular (aka orthogonal) eigenvectors.
 Q
Proof omitted.
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ENGG2013

12. Two sufficient conditions for diagonalizability
Distinct eigenvalues
Symmetric, skew-symmetric,
orthogonal
Diagonalizable
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ENGG2013

13. Example
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ENGG2013

14. Similarity
Definition: We say that two nn matrix A and B are similar if we can find an invertible matrix S such that
Example: and are similar,
The notion of diagonalization can be phrased in terms of similarity: matrix A is diagonalizable if and only if A is similar to a diagonal matrix.
\begin{bmatrix}
3 & 0\\
-1 & 1
\end{bmatrix}^{-1}
-1 & 0\\
2/3 & 1
\end{bmatrix}
\end{bmatrix} =
0 & 1
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15. More examples is similar to because and are similar. \begin{bmatrix}
1 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{bmatrix}^{-1}\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9
\end{bmatrix} \begin{bmatrix}
\end{bmatrix} = \begin{bmatrix}
1 & 3 & 2\\
7 & 9 & 8\\
4 & 6 & 5
\end{bmatrix}
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16. Application to conic sections
Ellipse : x2/a + y2/b = 1.
Hyperbola : x2/a – y2/b = 1.
Parabola y = ax2.
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ENGG2013

17. Application to conic sections
Is x2 – 4xy +2y2 = 1 a ellipse, or a hyperbola?
Rewrite using symmetric matrix
Find the characteristic polynomial
Solve for the eigenvalues
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18. Application to conic sections
Diagonalize
Change coordinates
\begin{bmatrix}
x & y
\end{bmatrix}\mathbf{P}^{-1} \begin{bmatrix}
1 & -2\\
-2 & 2
\end{bmatrix} \mathbf{P}
x\\y
\end{bmatrix} = 1
Hyperbola
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19. x2 – 4xy +2y2 = 1
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20. 2x2 + 2xy + 2y2 = 1 Rewrite using symmetric matrix
Compute the characteristic polynomial
Find the eigenvalues
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21. 2x2 + 2xy + 2y2 = 1
Columns of P are eigenvectors, normalized to norm 1.
Diagonalize
Change of variables
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22. 2x2 + 2xy + 2y2 = 1 v u
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