# Distinct Eigenvalues of a matrix

In one of the competitive exams, they asked the below question.

Q). The number of distinct eigenvalues of the matrix A is equal to ___

and The matrix A is given below,

    2 2 3 3

0 1 1 1

0 0 3 3

0 0 0 2


Since it is upper triangular matrix, the Eigenvalues are its diagonal elements. That is (1,3,2,2)

My answer: There are 4 Eigenvalues among which two are distinct (1,3) & two are same (2,2). So I answered 2 for that question.

Now the key answer came out from examination board & the answer is 3(distinct Eigenvalues are 1,3,2). That is there are 3 distinct Eigenvalues.

Is my way of understanding distinct correct or not? What is the correct answer?

• I must say, I don't understand your understanding. There are $3$ distinct eigenvalues: $1, 2, 3$. – Theo Bendit Feb 20 at 7:49

In mathematical English, the word "distinct" means that we count every object that occurs at least once, but we count each such object only once. For example, the number of distinct letters in the word "distinct" is $$6$$, since d, i, s, t, n, and c each occur in the word (and no other letters do); the number of distinct eigenvalues of $$A$$ is $$3$$, since $$1$$, $$2$$, and $$3$$ are the eigenvalues. This is the same as counting the number of elements in the sets {d,i,s,t,i,n,c,t} = {d,i,s,t,n,c} and $$\{2,1,3,2\}=\{1,2,3\}$$, respectively.
If we wanted to describe your interpretation, we could use "unrepeated" or "that appear exactly once" or "that appear with multiplicity $$1$$". For example, the number of letters that appear exactly once in the word "distinct" is $$4$$ (they are d, s, n, and c); the number of unrepeated eigenvalues of $$A$$ (the number of eigenvalues of $$A$$ that appear with multiplicity $$1$$) is $$2$$.