15: Diagonalizing Symmetric Matrices
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Symmetric matrices have many applications. For example, if we consider the shortest distance between pairs of important cities, we might get a table like this:
\[\begin{array}{c|ccc}& Davis & Seattle & San\; Francisco \\ \hline Davis & 0 & 2000 & 80 \\Seattle & 2000 & 0 & 2010 \\San \;Francisco & 80 & 2010 & 0\end{array}\]
Encoded as a matrix, we obtain:
\[M=\begin{pmatrix}0 & 2000 & 80 \\2000 & 0 & 2010 \\80 & 2010 & 0\end{pmatrix}=M^{T}.\]
Definition: symmetric Matrix
A matrix is symmetric if it obeys \[M=M^{T}.\]
One nice property of symmetric matrices is that they always have real eigenvalues. Review exercise 1 guides you through the general proof, but here's an example for \(2\times 2\) matrices:
Example 128
For a general symmetric \(2\times 2\) matrix, we have:
\begin{eqnarray*}P_{\lambda} \begin{pmatrix} a & b \\ b& d \end{pmatrix}&=&\det\begin{pmatrix}\lambda-a&-b\\-b&\lambda-d \end{pmatrix}\\&=& (\lambda-a)(\lambda-d)-b^{2} \\&=& \lambda^{2}-(a+d)\lambda-b^{2}+ad\\\Rightarrow \lambda &=& \frac{a+d}{2}\pm \sqrt{b^{2}+\left(\frac{a-d}{2}\right)^{2}}.\end{eqnarray*}
Notice that the discriminant \(4b^{2}+(a-d)^{2}\) is always positive, so that the eigenvalues must be real.
Now, suppose a symmetric matrix \(M\) has two distinct eigenvalues \(\lambda \neq \mu\) and eigenvectors \(x\) and \(y\):
\[Mx=\lambda x, \qquad My=\mu y.\]
Consider the dot product \(x\cdot y = x^{T}y = y^{T}x\) and calculate:
\begin{eqnarray*}x^{T}M y &=& x^{T}\mu y = \mu x\cdot y, \textit{ and }\\x^{T}M y &=& (y^{T}Mx)^{T} \textit{ (by transposing a \(1\times 1\) matrix)}\\&=& x^{T}M^{T}y \\&=& x^{T}My \\&=& x^{T}\lambda y \\&=& \lambda x\cdot y.\end{eqnarray*}
Subtracting these two results tells us that:
\begin{eqnarray*}0 &=& x^{T}My-x^{T}My=(\mu-\lambda)\,x\cdot y.\end{eqnarray*}
Since \(\mu\) and \(\lambda\) were assumed to be distinct eigenvalues, \(\lambda-\mu\) is non-zero, and so \(x\cdot y=0\). We have proved the following theorem.
Theorem
Eigenvectors of a symmetric matrix with distinct eigenvalues are orthogonal.
Example 129
The matrix \(M=\begin{pmatrix}2&1\\1&2\end{pmatrix}\) has eigenvalues determined by
\[\det(M-\lambda I)=(2-\lambda)^{2}-1=0.\]
So the eigenvalues of \(M\) are \(3\) and \(1\), and the associated eigenvectors turn out to be \(\begin{pmatrix}1\\1\end{pmatrix}\) and \(\begin{pmatrix}1\\-1\end{pmatrix}\). It is easily seen that these eigenvectors are orthogonal:
\[\begin{pmatrix}1\\1\end{pmatrix} \cdot \begin{pmatrix}1\\-1\end{pmatrix}=0\]
In chapter 14 we saw that the matrix \(P\) built from any orthonormal basis \((v_{1},\ldots, v_{n} )\) for \(\mathbb{R}^{n}\) as its columns,
\[P=\begin{pmatrix}v_{1} & \cdots & v_{n}\end{pmatrix}\, ,\]
was an orthogonal matrix:
\[P^{-1}=P^{T}, \textit{ or } PP^{T}=I=P^{T}P.\]
Moreover, given any (unit) vector \(x_{1}\), one can always find vectors \(x_{2}, \ldots, x_{n}\) such that \((x_{1},\ldots, x_{n})\) is an orthonormal basis. (Such a basis can be obtained using the Gram-Schmidt procedure.)
Now suppose \(M\) is a symmetric \(n\times n\) matrix and \(\lambda_{1}\) is an eigenvalue with eigenvector \(x_{1}\) (this is always the case because every matrix has at least one eigenvalue--see review problem 3). Let the square matrix of column vectors \(P\) be the following:
\[P=\begin{pmatrix}x_{1} & x_{2} & \cdots & x_{n}\end{pmatrix},\]
where \(x_{1}\) through \(x_{n}\) are orthonormal, and \(x_{1}\) is an eigenvector for \(M\), but the others are not necessarily eigenvectors for \(M\). Then
\[MP=\begin{pmatrix}\lambda_{1} x_{1} & Mx_{2} & \cdots & Mx_{n}\end{pmatrix}.\]
But \(P\) is an orthogonal matrix, so \(P^{-1}=P^{T}\). Then:
\begin{eqnarray*}P^{-1}=P^{T} &=& \begin{pmatrix}x_{1}^{T}\\ \vdots \\ x_{n}^{T}\end{pmatrix} \\\Rightarrow P^{T}MP &=& \begin{pmatrix}x_{1}^{T}\lambda_{1}x_{1} & * & \cdots & *\\x_{2}^{T}\lambda_{1}x_{1} & * & \cdots & *\\\vdots & & & \vdots\\x_{n}^{T}\lambda_{1}x_{1} & * & \cdots & *\\\end{pmatrix}\\&=& \begin{pmatrix}\lambda_{1} & * & \cdots & *\\0 & * & \cdots & *\\\vdots & * & & \vdots\\0 & * & \cdots & *\\\end{pmatrix}\\&=& \begin{pmatrix}\lambda_{1} & 0 & \cdots & 0\\0 & & & \\\vdots & & \hat{M} & \\0 & & & \\\end{pmatrix}\, .\\\end{eqnarray*}
The last equality follows since \(P^{T}MP\) is symmetric. The asterisks in the matrix are where “stuff'' happens; this extra information is denoted by \(\hat{M}\) in the final expression. We know nothing about \(\hat{M}\) except that it is an \((n-1)\times (n-1)\) matrix and that it is symmetric. But then, by finding an (unit) eigenvector for \(\hat{M}\), we could repeat this procedure successively. The end result would be a diagonal matrix with eigenvalues of \(M\) on the diagonal. Again, we have proved a theorem:
Theorem
Every symmetric matrix is similar to a diagonal matrix of its eigenvalues. In other words,
\[M=M^{T} \Leftrightarrow M=PDP^{T}\]
where \(P\) is an orthogonal matrix and \(D\) is a diagonal matrix whose entries are the eigenvalues of \(M\).
To diagonalize a real symmetric matrix, begin by building an orthogonal matrix from an orthonormal basis of eigenvectors:
Example 130
The symmetric matrix
\[M=\begin{pmatrix}2&1\\1&2\end{pmatrix}\, ,\]
has eigenvalues \(3\) and \(1\) with eigenvectors \(\begin{pmatrix}1\\1\end{pmatrix}\) and \(\begin{pmatrix}1\\-1\end{pmatrix}\) respectively. After normalizing these eigenvectors, we build the orthogonal matrix:
\[P = \begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}}\end{pmatrix}\, .\]
Notice that \(P^{T}P=I\). Then:
\[MP = \begin{pmatrix}\frac{3}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{3}{\sqrt{2}} & \frac{-1}{\sqrt{2}}\end{pmatrix} = \begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}}\end{pmatrix} \begin{pmatrix}3 & 0 \\0 & 1\end{pmatrix}.\]
In short, \(MP=DP\), so \(D=P^{T}MP\). Then \(D\) is the diagonalized form of \(M\) and \(P\) the associated change-of-basis matrix from the standard basis to the basis of eigenvectors.
Contributor
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)