7-3 Diagonal matrix and transpose

From the last chapter, we learned the way to compute the eigenvectors and eigenvalues of a matrix. We also learned the geometric meaning of eigenvectors and eigenvalues. You might say: Sure, here are the eigenvectors and eigenvalues of a matrix. So what? What is the application?

Don’t worry! You will find out the good stuff real soon. But for now, we have to learn one interesting type of matrix, which is related to the “good stuff” later.

The special matrix is called the diagonal matrix. The name “diagonal” comes from the fact that the matrix only has non-zero entry on the “diagonal” of the square matrix:

$$ A = \begin{bmatrix} x_{11} & 0 & 0 & \dots & 0 \\ 0 & x_{22} & 0 & \dots & 0 \\ 0 & 0 & x_{33} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & x_{nn} \end{bmatrix} $$

In convention, we call the entries that have the same column and row number “diagonal” terms. and the entries that have the different column and row number “off-diagonal” terms. For the diagonal matrix, it is usually a $n\times n$ matrix (we don’t consider the “rectangular” ones”). Also, all the “off-diagonal” terms must be zero. Notice that the diagonal terms don’t have to be non-zero.

Great, this matrix looks VERY simple! Indeed, it is very easy to work with. Let’s look at an interesting example:

Let’s say there is a matrix

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$

then what is $A^2$?

We can do a quick matrix multiplication to get the answer for that:

$$A^2 = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 + 6 & 2 + 8 \\ 3 + 12 & 6 + 16 \end{bmatrix} = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}$$

Well that was a breeze. But what if someone ask you, what is $A^{201}$ ? Then we are screwed. However, if you ask the same question on a diagonal matrix $D$ :

$$ D = \begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix} $$

the result of $D^{201}$ is simply:

$$ D^{201} = \begin{bmatrix} 1^{201} & 0\\ 0 & 2^{201} \end{bmatrix} $$

Not only that, if you tried to find the eigenvalue of a diagonal matrix, you will soon realize the characteristic polynomials of a diagonal matrix is extremely simple.

If take matrix

$$D = \begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix}$$

as a example. The characteristic polynomials is just:

$$det(D-\lambda I) = det(\begin{bmatrix} 1-\lambda & 0\\ 0 & 2-\lambda \end{bmatrix}) = (1-\lambda)(2-\lambda) = 0$$

This tells us the eigenvalue right away, $\lambda = 1$ or $\lambda = 2$. Which happens to be the diagonal terms of the matrix. Also, the eigenvectors of the matrix are easy too(Try to find them by yourself).

Alright! We now learned everything we need for the grand finale. In the next section, we are going to combine all the pieces and construct one of the most powerful weapons in linear algebra: Diagonalization.

7-3 Diagonal matrix and transpose

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