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So I'm a high school student currently studying the chapter matrices (which is interesting but confusing in its own way). And recently I came across a problem that involved passing a matrix through a quadratic function. I understood the squaring and adding part but there was a step which involved multiplying the constant with the identity matrix. My question is why do we do this? Sure, a constant just cannot be added to matrices as one is a scalar and the other a vector. But why do we take specifically the identity matrix to multiply with the constant? Why not, say, a matrix filled with 1s?

Again, I don't know much about matrices as I'm a beginner. Any simplified explanation would help. Thanks in advance!

Bernard
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StackExchange_User_4
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    Could you show us the quadratic function that you mentioned ? And the step that you'd like to understand ? – lmsteffan Sep 01 '20 at 09:25
  • The matrices of given dimensions are a vector space over the base field, and you can multiply any matrix with a scalar. The result is simply a matrix where each coefficient is multiplied by the scalar. – Bernard Sep 01 '20 at 09:31
  • @Imsteffan

    I don't have the book right now but I found a similar one on a website.

    https://people.richland.edu/james/lecture/m116/matrices/operations.html

    Scroll down until you find "Evaluating a function using a matrix"

    There i couldn't understand why they multiplied 3 with the identity matrix "I".

     – StackExchange_User_4 Sep 01 '20 at 12:19

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Hand-wavey, intuitive answer: It is multiples of the identity matrix that most act like scalars, both when multiplying with other matrices, and when using them to linearly transform vectors. So it is by far the most reasonable thing to do with the constant term as you insert a matrix into a polynomial.

Arthur
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  • So the identity matrix is for matrices is what 1 is for numbers. It doesn't change the value. Hmmm. Interesting. So its kiiiind of like converting a scalar into a matrix without changing its value (kinda)? But is there a more specific answer on how the practice came about? – StackExchange_User_4 Sep 05 '20 at 10:23