Linear algebraConcept reference

Diagonal matrices

A square diagonal matrix has zero entries off its main diagonal and multiplies each coordinate by its matching diagonal entry.

On this page 7 sections

01A square diagonal matrix has zeros off its main diagonal#

A square diagonal matrix has zeros off its main diagonal. Each diagonal entry scales one input coordinate.

For $D=\operatorname{diag}(2,-1)$ and $x=(3,-2)$ , the first output is $2(3)=6$ . The second is $(-1)(-2)=2$ .

The output column vector is

Dx=(6,2)

Here and throughout this lesson, tuples list column-vector coordinates.

For $D=\operatorname{diag}(d_1,d_2)$ , each factor stays with its own coordinate.

Coordinate by coordinate,

Dx=(d_1x_1,d_2x_2)

A zero factor erases that coordinate. A negative factor reverses its sign.

Check your reasoning

$D=\operatorname{diag}(3,-2)$ and $x=(-1,4)$ . What is the second coordinate of $Dx$ ?

Show answer and explanation

-8

The second factor is $-2$ : $(-2)(4)=-8$ .

$D=\operatorname{diag}(2,3)$ sends $(1,1)$ to $(2,3)$ . The first coordinate doubles; the second triples.

The equal factors in $\operatorname{diag}(2,2)$ would instead double both coordinates. A diagonal matrix need not use one common multiplier.

Check your reasoning

A filter doubles channel 1 and silences channel 2. Choose its diagonal matrix.

Show answer and explanation

$\operatorname{diag}(2,0)$

Use factors 2 and 0, in that order.

Check your reasoning

$D=\operatorname{diag}(-1,3)$ sends $(2,1)$ to $(-2,1)$ , a learner says. What needs fixing?

Show answer and explanation

Second entry must be 3.

The second entry also scales: $3(1)=3$ .

Key takeaway