Representing Square Matrices with Constant and Identity Matrix - A Guide

[pjunky]

Is is possible to represent a "square matrix" with the product of the constant and identity matrix of same order of given matrix


EX:-A=[ ] 4x4 matrix

can I make it something like this A=(K) I 4x4

K= constant
I=identity matrix

 

[HallsofIvy]

Is is possible to represent a "square matrix" with the product of the constant and identity matrix of same order of given matrix


EX:-A=[ ] 4x4 matrix

can I make it something like this A=(K) I 4x4

K= constant
I=identity matrix

It's not clear what you are asking. If I is the identity matrix, then (K)I= K. But what doe you mean by the "constant" matrix? A= (K) I= K only if A is already the "constant" matrix itself.

 

[Tac-Tics]

No. There are tons of trivial counter examples. Any square matrix with components that are *not* along the diagonal.

 

[pjunky]

It's not clear what you are asking. If I is the identity matrix, then (K)I= K. But what doe you mean by the "constant" matrix? A= (K) I= K only if A is already the "constant" matrix itself.

for example
if the matrix A is some thing like this:-
2 0 0
0 2 0
0 0 2 =====> A=(2)I
where K=2
I=identity matrix of order 3



Now what I want to know is if matrix A is
a b c
d e f
g h i =======>A=(k)I

is it possible to shrink A in this form for a square matrix
how can I find what exactly k is??

 

[statdad]

for example
if the matrix A is some thing like this:-
2 0 0
0 2 0
0 0 2 =====> A=(2)I
where K=2
I=identity matrix of order 3



Now what I want to know is if matrix A is
a b c
d e f
g h i =======>A=(k)I

is it possible to shrink A in this form for a square matrix
how can I find what exactly k is??

You can't do what you want if the matrix $$A$$ does not have a form like these:

$$\begin{bmatrix} 4 & 0\\0 & 4 \end{bmatrix}, \quad \begin{bmatrix} -3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3 \end{bmatrix}$$

So, even more directly, if you start with

$$\begin{bmatrix} 4 & 2\\-8 & \pi \end{bmatrix}$$

you cannot write this as $$k I_2$$ no matter how imaginative you are in selecting the number $$k$$.

 

[statdad]

I must add - this is the point both HallsOfIvy and Tac-Tics were making.

 

[pjunky]

yeah I got the point
@ all people thanks for your help

 

[chota]

so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?

 

[HallsofIvy]

so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?

Have you actually tried this multiplication? The number k times I is exactly a matrix with "k" along the diagaonal and zeros everywhere else. Why are you even asking such a question? It's a lot like asking repeatedly if 1+ 1= 3. DO it and see for yourself!

 

[statdad]

so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?

This is an indubitable fact of mathematics.