Eigenvectors and Eigenvalues

Eigen Schuld = "Your own" fault (you numbskull!)

Matrix multiplication and eigenvectors

Given a set of basis vectors $\{\ket 1, \ket 2, \ket 3...\}$, we can represent any state vector, $\ket \alpha$ as a linear superposition of the basis vectors: $$\ket \alpha=a_1\ket 1+a_2\ket 2+a_3\ket 3+...$$ where the components are in general complex numbers.

So, a short hand way of representing a specific ket is to write its components out as a column vector: $$\ket{\alpha} \rightarrow \mym{a}= \begincv a_1 \\ a_2 \\ a_3 \\ . \\ . \endcv $$

The bra $\bra \alpha$ corresponding to a particular ket is the Hermitian conjugate of $\ket \alpha$, written as a row vector $$\bra{\alpha}\equiv \begincv a_1^* & a_2^* & a_3^* &.&. \endcv$$

We've seen how to calculate the dot product of a row vector with a column vector, represented by the inner product: $$\innerp{\beta}{\alpha}=b_1^*a_1+b_2^*a_2+b_3^*a_3+....$$

Matrix multiplication

Now we're going to do a bit more with matrix multiplication to represent operators which in turn represent the act of quantum mechanical measurement.

In linear algebra, an $N$-dimensional matrix $\mym M$ is written like this... $$\mym M=\begincv M_{11} & M_{12} &...&M_{1N}\\ M_{21} & M_{22} &...&M_{2N}\\ .&.&&.\\ M_{N1} & M_{N2} &...&M_{NN} \endcv $$ So, $M_{ij}$ refers to the number in row $i$ and column $j$ of the matrix.

Multiplying a matrix by a column vector gives another column vector: $$ \begincv 11 & 12 \\ 21 & 22 \\ \endcv \begincv 3 \\ 5 \endcv=\begincv 11*3+12*5 \\ 21*3+22*5\endcv =\begincv 99 \\ 173\endcv.$$ If we write this, in matrix notation, as $\mym M\mym a=\mym b$, we can see that the element in row $i$ of the column vector answer is given by... $$\sum_j M_{ij}a_j=b_i.$$

Eigenvectors

Linear dependence
Two vectors are linearly dependent if one vector is a scalar multiple of the other.

"Linearly dependent" for good ol' real vectors in space $\myv a$ and $\myv b$ means that $\myv a \parallel \myv b$. They're parallel.

  1. Are our two vectors $\mym a=\begincv 3\\5\endcv$ and $\mym b= \begincv 99 \\ 173\endcv$ linearly dependent or not? (If *not*, they are linearly independent.)
  2. Choose a value for the complex scalar number $w$ such that the two vectors below will be linearly dependent: $$\mym a = \begincv 3 \\ 1+2i \endcv;\ \ \ \mym b = \begincv 3-3i\\ w\endcv.$$

As you might imagine, as with our original example of matrix multiplication, most of the time if multiply some matrix $\mym M$ by some arbitrary column vector $\mym a$, you will end up with a column vector $\mym b$ which is linearly independent of $\mym a$.

But, if you stumble on one which is "parallel"...

If $\mym a$ satisfies the "eigenvalue equation": $$\mym M\mym a = \lambda \mym a,$$ where $\lambda$ is a scalar constant, then we say:

  1. $\mym a$ is an eigenvector of the matrix $\mym M$,
  2. $\lambda$ is the eigenvalue which corresponds to $\mym a$.

Consider the matrix and column vectors $$\mym M = \begincv 2 & 1 \\ 0 & 1 \\ \endcv; \ \ \mym a = \begincv 1 \\ 1 \endcv; \ \ \mym b=\begincv 2 \\ 0 \endcv.$$ Is either $\mym a$ or $\mym b$ an eigenvector of $\mym M$? If so, what is the corresponding eigenvalue?

Finding eigenvalues, eigenvectors in Sagemath

Eigenvectors.pdf has examples of things we want to do with matrices.

Homework

Read all of section 2.1.*, and sections 2.2.1 (Spin component in a general direction) and 2.2.2 (Hermitian operators).

Write down your questions, and bring those to class.

Revised In Chapter 2, do problems 1, 2, 3 in SageMath (as much as possible) and bring a printout of your notebook file to class.