Matrices | Bsc 2nd Semester | BA Bsc | Engineering Mathematics | Lecture -2 | Krishna Publication |
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This video Contains -
00:00 - Lecture Intro
00:54 - Conjugate of a Matrix
03:06 - Transposed Conjugate of a Matrix
04:45 - Hermitian Matrix
06:00 - Skew Hermitian Matrix
07:57 - If A is a Hermitian matrix then show that iA is skew Hermitian.
10:04 - If A and B are symmetric matrices , then show that AB is symmetric if and only if A and B commutate. i.e. AB=BA.
12:52 - If A be any matrix, then prove that AA’ and A’A are both symmetric matrices.
14:04 - Show that the matrix B’AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.
16:58 - Show that the matrix
[■8(𝑖&3+2𝑖&−2−𝑖@−3+2𝑖&0&3−4𝑖@2−𝑖&−3−4𝑖&−2𝑖)] is skew- Hermitian.
19:20 - If A =[■8(3&2−3𝑖&3+5𝑖@2+3𝑖&5&𝑖@3−5𝑖&−𝑖&7)], Prove that A is a Hermitian matrix.
21:14 - Express [■8(−2+3𝑖&1−𝑖&2+𝑖@3&4−5𝑖&5@1&1+𝑖&−2+2𝑖)] as the sum of a Hermitian and a skew-Hermitian matrix.
26:20 Express the following matrix as the sum of symmetric and skew-symmetric matrix :
[■8(1&2&4@−2&5&3@−1&6&3)]
28:41 Show that every square matrix is uniquely expressible as the sum of a symmetric matrix and a skew symmetric matrix.
29:24 Minor
30:43 Cofactor
34:10 Unitary Matrix
34:35 If A , B be n-rowed unitary matrices, AB and BA are also unitary matrices.
35:57 Prove that 𝐵=1/√3 [■8(1&1+𝑖@1−𝑖&−1)] is unitary.
38:05 Show that the matrix A=[■8(𝛼+𝑖𝛾&−𝛽+𝑖𝛿@𝛽+𝑖𝛿&𝛼−𝑖𝛾)] is a unitary matrix, if 𝛼^2+𝛽^2+𝛾^2+𝛿^2=1.
42:51 If A is a unitary matrix, show that 𝐴^(−1) is also unitary.
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Matrices | Bsc 2nd Semester | BA Bsc | Engineering Mathematics | Lecture -1 | Krishna Publication