Transposing a vector

$ \left[ \begin{matrix} 1 \\ 0 \\ 2 \\ 5 \\ -2 \end{matrix} \right]^T = \left[ \begin{matrix} 1 \quad 0 \quad 2 \quad 5 -2\end{matrix} \right]$

$ \left[ \begin{matrix} 1 \quad 0 \quad 2 \quad 5 -2\end{matrix} \right]^T = \left[ \begin{matrix} 1 \\ 0 \\ 2 \\ 5 \\ -2 \end{matrix} \right]$

$V^{TT}=V$

Transposing a matrix

$ \left[ \begin{matrix} 1 \quad 5\\ 0 \quad 6 \\ 2 \quad 8 \\ 5 \quad 3 \\ -2 \quad 0 \end{matrix} \right]^T = \left[ \begin{matrix} 1 \quad 0 \quad 2 \quad 5 \quad -2 \\ 5 \quad 6 \quad 8 \quad 3 \quad 0 \end{matrix} \right]$

In [1]:
import numpy as np
import matplotlib.pyplot as plt
In [2]:
r = np.random.randn(1,10)

rt1 = np.transpose(r)
rt2 = r.T

print(np.shape(r))
print(np.shape(rt1))
print(np.shape(rt2))

(1, 10)
(10, 1)
(10, 1)
In [3]:
mat = np.random.randn(8,4)
matT = mat.T

fig,ax = plt.subplots(1,2)
ax[0].imshow(mat)
ax[0].set_title('M')

ax[1].imshow(matT)
ax[1].set_title('$M^T$')

for i in ax:
    i.set_xticks([])
    i.set_yticks([])

plt.show()

Symmetric matrices

$ \left[ \begin{matrix} \quad 1 \quad -1 \quad \quad 0\\ -1 \quad -2 \quad -4 \\ \quad 0 \quad -4 \quad \quad 5 \end{matrix} \right]$

$\text {Symmetric}: A = A^T$

In [4]:
m = 4
n = 9

amat = np.random.randn(m,n)

# palindrome: 回文
amatama = amat@amat.T

# show that this is square
print(np.shape(amatama))

# show that this is symmetric
print(amatama - amatama.T )

(4, 4)
[[0. 0. 0. 0.]
 [0. 0. 0. 0.]
 [0. 0. 0. 0.]
 [0. 0. 0. 0.]]