I have read that, for
$$I - \alpha P$$
where $I$ is the $n\times n$ identity matrix, $\alpha \in (0,1]$, and $P$ is the transition probability matrix with dimensions $n \times n$, $I - \alpha P$ is an invertible matrix because the eigenvalues of any transition probability matrix lie within the unit circle of the complex plane. "Thus no eigenvalue of $\alpha P$ can be equal to 1, which is a necessary and sufficient condition for $I - \alpha P$ to be invertible."

Why is that?


Question: Java - Is the Point class's hashCode() method any good, or should I override it and write my own?
Answer: If you're searching for the hashCode() of java.awt.Point, it is defined in java.awt.geom.Point2D.

/**
 * Returns the hashcode for this &lt;code&gt;Point2D&lt;/code&gt;.
 * @return      a hash code for this &lt;code&gt;Point2D&lt;/code&gt;.
 */
public int hashCode() {
    long bits = java.lang.Double.doubleToLongBits(getX());
    bits ^= java.lang.Double.doubleToLongBits(getY()) * 31;
    return (((int) bits) ^ ((int) (bits &gt;&gt; 32)));
}


Note that the question "Will it hash well?" is hard to answer, it depends primarily on usage pattern.

You can access the source code of nearly all "standard Java classes", just search for the src.zip file in your JDK installation directory (or use an IDE like Eclipse/NetBeans and click F3 on the class name).