Please explain this concept of Linear algebra ( Matrices and Vector ) ?

Case 1 : When Matrix is multiplied with a Column Vector ie #Ax#
Case 2 : When a Scalar is multiplied to a Column Vector ie #lambdax#

If #Ax = lambdax# ;

Here the Matrix #A# and the scalar #lambda# are not equal (bcoz matrix and scalar are totally different things) then how their products with vector #x# is equal ??

!!! plz give some nice explanation !!!

1 Answer
Apr 3, 2018

See below.

Explanation:

The basic rule you need to understand is that when you multiply two matrices #A# and #B# you will obtain a third matrix #C# which is possibly different in size from both #A# and #B#.

The rule states that, if #A# is a #(n \times m)# matrix and #B# is a #(m \times p)# matrix, then #C# will be a #(n \times p)# matrix (note that the number of columns of #A# and the number of rows of #B# must be the same, in this case #m#, otherwise you can't multiply #A# and #B#).

Also, you can consider vectors as special matrices, having only one row (or column).

Let's say that in your case #A# is a #(n \times n)# matrix. It follows that #x# must be a column vector with #n# rows and one column. So, by the rule above, the product between #A# and #x# is of the form

#(n\times n)(n\times 1) = (n\times 1)#

And thus #Ax# has the same shape of #x# itself.

In the same way, #\lambda x# is just #x# multiplied by some constant, and thus its shape won't change.

So, being both vectors of the same shape #(n \times 1)#, it makes sense to ask if they are equal.

P.S. Note that it is necessary for #A# to be a square matrix. In fact, if #A# is a #(m \times n)# matrix, then #Ax# is a #(m\ times 1)# vector, and can't be a multiple of #x#.