Final answer:
The student asked about matrix properties and spectra but provided data related to statistical distributions. The information given pertains to skewness, where symmetrical data mean the mean, median, and mode coincide; skewed data means the mean differs from the median; and bimodal distribution has two modes. To answer accurately about matrices, we would need the matrices themselves.
Step-by-step explanation:
The subject matter of the question involves understanding and classifying matrices based on their properties (symmetric, skew-symmetric, orthogonal) and finding their spectra (eigenvalues). However, the provided information appears to relate to the concepts of skewness and the typical relationships between the mean, median, and mode in a data set, which is part of statistics, not matrix theory. Therefore, there's a mismatch between the question asked and the information provided, making it difficult to connect Theorems 1 and 5 to the context. Below is how we would analyze the data in terms of its distribution:
- Symmetrical distribution: When the data are evenly distributed on both sides of a central value, meaning the mean, median, and mode all coincide.
- Skewed to the left (negative skewness): The data are stretched out more to the left of the central value, and typically, the mean is less than the median.
- Skewed to the right (positive skewness): The data are stretched out more to the right of the central value, and typically, the mean is greater than the median.
- A distribution with two modes is described as bimodal.
Without the actual matrices, we cannot determine whether they are symmetric, skew-symmetric, or orthogonal, nor can we calculate their spectra.