Most generic eigenvalues of a \(3\times 3\) traceless matrix with zero determinant are given as (\(a\) is some constant) :
\(a, -a, 0\)
For any square matrix, the sum of the eigenvalues equals the trace and the product of the eigenvalues equals the determinant.
Let the eigenvalues be \(\lambda_{1},\lambda_{2},\lambda_{3}\). Traceless means \(\lambda_{1}+\lambda_{2}+\lambda_{3}=0\), and zero determinant means \(\lambda_{1}\lambda_{2}\lambda_{3}=0\).
The product condition forces at least one eigenvalue to vanish, say \(\lambda_{3}=0\). The trace condition then gives \(\lambda_{1}+\lambda_{2}=0\), so \(\lambda_{2}=-\lambda_{1}\).
Writing \(\lambda_{1}=a\), the most general set is \(a, -a, 0\).
The set \(0, 0, 0\) is only the special case \(a=0\); \(0, 0, a\) has trace \(a\), which is non-zero unless \(a=0\); and \(a, -2a, a\) has determinant \(-2a^{3}\), non-zero unless \(a=0\).
Check: \(a+(-a)+0=0\) (trace) and \(a\times(-a)\times 0=0\) (determinant), so both conditions hold for every value of \(a\).
Hence, \(a, -a, 0\).
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Important Questions from Mixed Topic (MH-SET)
.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
