ENGG. MATRICES LEC 2 | FUNDAMENTALS OF ENGINEERING MATHS | ALL BRANCHES | DINESH SIR
DINESH SIR Live Study・15 minutes read
Matrices can be expressed as a sum of symmetric and skew symmetric parts by using specific formulas and calculations. The process involves finding the symmetric matrix (p) and the skew symmetric matrix (q), with practical examples provided to illustrate the concept and properties of these matrices.
Insights
- Any square matrix can be broken down into a sum of symmetric and skew symmetric matrices, showcasing the versatility and compositional nature of matrices in linear algebra.
- The process of converting matrices into Hermitian and skew Hermitian forms involves specific formulas and calculations, demonstrating a structured approach to transforming matrices and understanding their unique properties in complex analysis.
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Recent questions
What are symmetric and skew symmetric matrices?
Symmetric matrices have elements mirrored across the main diagonal, while skew symmetric matrices have elements that are negatives of each other across the main diagonal.
How can a square matrix be expressed as symmetric and skew symmetric matrices?
Any square matrix can be expressed as the sum of a symmetric matrix and a skew symmetric matrix.
What is the formula to calculate a symmetric matrix?
The formula to calculate a symmetric matrix is 1/2(a + a transpose), where 'a' is the original matrix.
How do you find the skew symmetric matrix of a square matrix?
The skew symmetric matrix can be found using the formula 1/2(a - a transpose), where 'a' is the original matrix.
What are the properties of Hermitian and skew Hermitian matrices?
Hermitian matrices have elements that are equal to their complex conjugates, while skew Hermitian matrices have elements that are negatives of their complex conjugates.
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Summary
00:00
Matrix Decomposition: Symmetric and Skew-Symmetric Calculations
- Introduction to the concept of matrices, specifically focusing on symmetric and skew symmetric matrices.
- Explanation of how any square matrix can be expressed as a sum of symmetric and skew symmetric matrices.
- Detailed instructions on how to calculate the symmetric matrix (p) using the formula 1/2(a + a transpose).
- Step-by-step process of finding the skew symmetric matrix (q) using the formula 1/2(a - a transpose).
- Practical example provided with a given matrix to demonstrate the calculation of p and q.
- Clarification on the characteristics of symmetric and skew symmetric matrices, emphasizing the diagonal elements and their signs.
- Application of the formulas to determine the values of p and q for a specific matrix.
- Further explanation of the properties of Hermitian and skew Hermitian matrices, highlighting their unique representation as a sum.
- Detailed instructions on calculating the Hermitian matrix (p) and skew Hermitian matrix (q) using specific formulas.
- Practical example given with a matrix to illustrate the process of finding p and q for Hermitian and skew Hermitian matrices.
16:32
Matrix Transformations: Hermitian and Symmetric Matrices
- Matrix A consists of the values 2+3i, 0, 4i, 5i, 8, and 5i, 8, 1-i, -3, 1, 3, 6. By performing calculations, it is determined that the value of A is 11, with specific steps outlined for each computation.
- The process of converting a given square matrix into a Hermitian matrix is explained, with detailed steps provided for the transformation. The resulting Hermitian matrix is then further analyzed to showcase its properties.
- The concept of converting any matrix into a symmetric plus skew-symmetric matrix is introduced, highlighting the ability to transform matrices into different forms based on specific properties. The importance of understanding these transformations is emphasized for comprehensive learning.
