import numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as mpatches\nfrom matplotlib.patches import FancyArrowPatch\nfrom matplotlib.gridspec import GridSpec\n\n# \u2500\u2500 shared helpers \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\ndef draw_grid(ax, A=None, color='lightgray', alpha=0.5, lw=0.8, extent=3):\n """Draw a transformed (or standard) grid."""\n lines = np.linspace(-extent, extent, 2 * extent + 1)\n for v in lines:\n pts_h = np.array([[-extent, v], [extent, v]]).T # horizontal line\n pts_v = np.array([[v, -extent], [v, extent]]).T # vertical line\n if A is not None:\n pts_h = A @ pts_h\n pts_v = A @ pts_v\n ax.plot(*pts_h, color=color, alpha=alpha, lw=lw, zorder=0)\n ax.plot(*pts_v, color=color, alpha=alpha, lw=lw, zorder=0)\n\ndef draw_arrow(ax, vec, origin=None, color='black', lw=2, label=None, label_offset=None):\n if origin is None:\n origin = np.zeros(2)\n ax.annotate('', xy=origin + vec, xytext=origin,\n arrowprops=dict(arrowstyle='->', color=color, lw=lw))\n if label:\n pos = origin + vec + (label_offset if label_offset is not None else np.array([0.1, 0.1]))\n ax.text(*pos, label, color=color, fontsize=12)\n\ndef style_ax(ax, title='', xlim=(-3.5, 3.5), ylim=(-3.5, 3.5)):\n ax.set_xlim(*xlim); ax.set_ylim(*ylim)\n ax.set_aspect('equal')\n ax.axhline(0, color='gray', lw=0.6); ax.axvline(0, color='gray', lw=0.6)\n ax.set_title(title, fontsize=11, pad=8)\n ax.set_xlabel('$x_1$'); ax.set_ylabel('$x_2$')<\/code><\/pre>\n\n\n\n1. Active vs. Passive Interpretation<\/h2>\n\n\n\n
Given a matrix $A$ and a vector $v$, the product $Av$ has two valid readings:<\/p>\n\n\n\n
Active<\/strong> \u2014 $A$ moves<\/em> $v$ to a new location in the same coordinate system.
Passive<\/strong> \u2014 $A$ redefines the rulers<\/em>; the vector stays put but its coordinates change.<\/p>\n\n\n\nBoth produce the same numbers. The distinction is conceptual: are the vectors moving, or is the coordinate system?<\/p>\n\n\n\n
Below we show both views of the same matrix $A = \\begin{bmatrix} 2 & 0.5 \\ 0.5 & 1.5 \\end{bmatrix}$ acting on $v = [1, 1]^\\top$.<\/p>\n\n\n\n
A = np.array([[2.0, 0.5],\n [0.5, 1.5]])\nv = np.array([1.0, 1.0])\nAv = A @ v\n\nfig, axes = plt.subplots(1, 2, figsize=(13, 5))\n\n# \u2500\u2500 Active view \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\nax = axes[0]\ndraw_grid(ax, color='lightgray')\ndraw_arrow(ax, v, color='steelblue', lw=2.5, label='$v$', label_offset=np.array([ 0.05, 0.15]))\ndraw_arrow(ax, Av, color='crimson', lw=2.5, label='$Av$', label_offset=np.array([ 0.1, -0.3]))\nstyle_ax(ax, title='Active view \u2014 $A$ moves $v$ to $Av$\\n(coordinate system fixed)')\n\n# \u2500\u2500 Passive view \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\nax = axes[1]\ndraw_grid(ax, color='lightgray') # original grid (faint)\ndraw_grid(ax, A=A, color='#90c4e4', alpha=0.7, lw=1.2) # transformed grid\ndraw_arrow(ax, v, color='steelblue', lw=2.5, label='$v$', label_offset=np.array([0.05, 0.15]))\nstyle_ax(ax, title='Passive view \u2014 $A$ reshapes the coordinate grid\\n(vector $v$ stays fixed)')\nhandles = [\n mpatches.Patch(facecolor='lightgray', label='Original grid'),\n mpatches.Patch(facecolor='#90c4e4', label='Transformed grid'),\n]\naxes[1].legend(handles=handles, loc='lower right', fontsize=9)\n\nfig.suptitle('Two interpretations of the same matrix $A$', fontsize=13, y=1.01)\nplt.tight_layout()\nplt.show()<\/code><\/pre>\n\n\n\n![\"\"](\"https:\/\/tensorzen.online\/wp-content\/uploads\/2026\/06\/image-10-1024x489.png\")
<\/figure>\n\n\n\n2. Columns of $A$ as Images of Basis Vectors<\/h2>\n\n\n\n
The most useful geometric fact:<\/p>\n\n\n\n
\nThe $j$-th column of $A$ is exactly where the $j$-th standard basis vector $e_j$ lands.<\/strong><\/p>\n<\/blockquote>\n\n\n\nThis is because $A e_j$ selects the $j$-th column. So the matrix is completely described by where it sends the axes:<\/p>\n\n\n\n
$$A = \\begin{bmatrix} | & | \\ Ae_1 & Ae_2 \\ | & | \\end{bmatrix}$$<\/p>\n\n\n\n
Everything else follows by linearity \u2014 any vector $v = v_1 e_1 + v_2 e_2$ maps to $Av = v_1(Ae_1) + v_2(Ae_2)$.<\/p>\n\n\n\n
e1 = np.array([1.0, 0.0])\ne2 = np.array([0.0, 1.0])\nAe1, Ae2 = A @ e1, A @ e2\n\nprint(f"A =\\n{A}")\nprint(f"\\nAe1 = {Ae1} \u2190 first column of A")\nprint(f"Ae2 = {Ae2} \u2190 second column of A")\n\n# Verify linearity: A @ v = v[0]*Ae1 + v[1]*Ae2\nv_reconstructed = v[0] * Ae1 + v[1] * Ae2\nprint(f"\\nv = {v}")\nprint(f"Av via matrix multiply : {A @ v}")\nprint(f"Av via linear combo : {v_reconstructed} \u2713")<\/code><\/pre>\n\n\n\nA =\n[[2. 0.5]\n [0.5 1.5]]\n\nAe1 = [2. 0.5] \u2190 first column of A\nAe2 = [0.5 1.5] \u2190 second column of A\n\nv = [1. 1.]\nAv via matrix multiply : [2.5 2. ]\nAv via linear combo : [2.5 2. ] \u2713<\/pre>\n\n\n\nfig, axes = plt.subplots(1, 2, figsize=(13, 5))\n\nfor ax, (title, draw_A) in zip(axes, [\n ('Before: standard basis', False),\n ('After: columns of $A$ are new basis', True)\n]):\n draw_grid(ax, A=(A if draw_A else None), color='lightgray')\n if draw_A:\n draw_arrow(ax, Ae1, color='crimson', lw=2.5, label='$Ae_1$ (col 1)', label_offset=np.array([0.1, -0.35]))\n draw_arrow(ax, Ae2, color='darkorange', lw=2.5, label='$Ae_2$ (col 2)', label_offset=np.array([0.1, 0.1]))\n # show transformed v\n draw_arrow(ax, A @ v, color='purple', lw=2, label='$Av$', label_offset=np.array([0.1, 0.1]))\n else:\n draw_arrow(ax, e1, color='crimson', lw=2.5, label='$e_1$', label_offset=np.array([ 0.05, -0.25]))\n draw_arrow(ax, e2, color='darkorange', lw=2.5, label='$e_2$', label_offset=np.array([-0.35, 0.1]))\n draw_arrow(ax, v, color='purple', lw=2, label='$v$', label_offset=np.array([ 0.1, 0.1]))\n style_ax(ax, title=title, xlim=(-3.5, 3.5), ylim=(-2, 4))\n\nfig.suptitle('Columns of $A$ are the images of $e_1$ and $e_2$', fontsize=13, y=1.01)\nplt.tight_layout()\nplt.show()<\/code><\/pre>\n\n\n\n![\"\"](\"https:\/\/tensorzen.online\/wp-content\/uploads\/2026\/06\/image-11-1024x453.png\")
<\/figure>\n\n\n\n3. Non-Singular vs. Singular: Geometric Meaning<\/h2>\n\n\n\n
A matrix is non-singular<\/strong> (invertible) when its columns are linearly independent \u2014 they still span the full space after transformation. A singular<\/strong> matrix collapses at least one dimension.<\/p>\n\n\n\nThe determinant<\/strong> measures the signed area (2D) or volume (nD) scaling factor:<\/p>\n\n\n\n$$\\det(A) \\neq 0 \\iff \\text{non-singular} \\iff \\text{no dimension is collapsed}$$<\/p>\n\n\n\n
Below we compare three matrices:<\/p>\n\n\n\n
\n- $A_1$ \u2014 non-singular, stretches and rotates<\/li>\n\n\n\n
- $A_2$ \u2014 singular, collapses 2D onto a line<\/li>\n\n\n\n
- $A_3$ \u2014 non-singular, reflection (det < 0, orientation flipped)<\/li>\n<\/ul>\n\n\n\n
# Unit square corners (and closing point)\nsquare = np.array([[0,1,1,0,0],\n [0,0,1,1,0]], dtype=float)\n\nA1 = np.array([[2.0, 0.5], # non-singular stretch\n [0.5, 1.5]])\nA2 = np.array([[1.0, 2.0], # singular: col2 = 2*col1\n [0.5, 1.0]])\nA3 = np.array([[-1.0, 0.0], # reflection across y-axis\n [ 0.0, 1.0]])\n\nmatrices = [(A1, 'Non-singular\\n$A_1$, det={:.2f}'.format(np.linalg.det(A1))),\n (A2, 'Singular\\n$A_2$, det={:.2f}'.format(np.linalg.det(A2))),\n (A3, 'Non-singular (reflection)\\n$A_3$, det={:.2f}'.format(np.linalg.det(A3)))]\n\nfig, axes = plt.subplots(2, 3, figsize=(14, 9))\n\nfor col, (M, label) in enumerate(matrices):\n for row, (title_sfx, use_M) in enumerate([('Before', False), ('After', True)]):\n ax = axes[row][col]\n draw_grid(ax, A=(M if use_M else None))\n sq = M @ square if use_M else square\n ax.fill(sq[0], sq[1], alpha=0.25, color='steelblue')\n ax.plot(sq[0], sq[1], color='steelblue', lw=2)\n t = label if use_M else 'Standard basis'\n style_ax(ax, title=t, xlim=(-4,4), ylim=(-3,4))\n if use_M:\n c1, c2 = M[:, 0], M[:, 1]\n draw_arrow(ax, c1, color='crimson', lw=2, label='col 1', label_offset=np.array([ 0.1, -0.3]))\n draw_arrow(ax, c2, color='darkorange', lw=2, label='col 2', label_offset=np.array([ 0.1, 0.1]))\n\naxes[0][0].set_ylabel('Before', fontsize=12)\naxes[1][0].set_ylabel('After $A$', fontsize=12)\nfig.suptitle('Effect of non-singular vs. singular matrices on the unit square', fontsize=13)\nplt.tight_layout()\nplt.show()<\/code><\/pre>\n\n\n\n![\"\"](\"https:\/\/tensorzen.online\/wp-content\/uploads\/2026\/06\/image-12-1024x652.png\")
<\/figure>\n\n\n\nObservations:<\/strong><\/p>\n\n\n\n\n- $A_1$ (non-singular): the square becomes a parallelogram \u2014 area changes but it remains 2D<\/li>\n\n\n\n
- $A_2$ (singular): the square collapses to a line segment<\/strong> \u2014 all area is destroyed, the transformation is irreversible<\/li>\n\n\n\n
- $A_3$ (reflection): the square flips \u2014 the negative determinant signals the orientation reversal<\/strong><\/li>\n<\/ul>\n\n\n\n
4. Determinant as a Volume Scaling Factor<\/h2>\n\n\n\n
The determinant is not just a number \u2014 it is the signed area scaling factor<\/strong> of the transformation.<\/p>\n\n\n\n$$\\text{Area after} = |\\det(A)| \\times \\text{Area before}$$<\/p>\n\n\n\n
The sign encodes orientation: positive means the transformation preserves orientation (no reflection), negative means it reverses it.<\/p>\n\n\n\n
unit_area = 1.0 # area of unit square\n\nfor M, name in [(A1,'A1 (non-singular)'), (A2,'A2 (singular)'), (A3,'A3 (reflection)')]:\n d = np.linalg.det(M)\n print(f"{name}: det = {d:+.4f}, area after = {abs(d):.4f}, orientation {'preserved' if d > 0 else 'reversed' if d < 0 else 'destroyed'}")<\/code><\/pre>\n\n\n\nA1 (non-singular): det = +2.7500, area after = 2.7500, orientation preserved\nA2 (singular): det = +0.0000, area after = 0.0000, orientation destroyed\nA3 (reflection): det = -1.0000, area after = 1.0000, orientation reversed<\/pre>\n\n\n\n5. SVD Decomposes the Transformation into Three Steps<\/h2>\n\n\n\n
The SVD $A = U \\Sigma V^\\top$ breaks every matrix transformation into three interpretable steps:<\/p>\n\n\n\n
$$v \\;\\xrightarrow{V^\\top}\\; \\text{rotate\/reflect} \\;\\xrightarrow{\\Sigma}\\; \\text{scale along axes} \\;\\xrightarrow{U}\\; \\text{rotate\/reflect}$$<\/p>\n\n\n\n
\n- $V^\\top$ \u2014 rotates the input to align with the matrix’s natural input directions<\/strong> (right singular vectors)<\/li>\n\n\n\n
- $\\Sigma$ \u2014 stretches or compresses each axis independently by the singular values<\/strong> $\\sigma_i$<\/li>\n\n\n\n
- $U$ \u2014 rotates the result into the output orientation<\/strong> (left singular vectors)<\/li>\n<\/ul>\n\n\n\n
Non-singular means all $\\sigma_i > 0$ \u2014 every axis is scaled by a nonzero amount, so no dimension is lost.<\/p>\n\n\n\n
U, s, Vt = np.linalg.svd(A1)\nSigma = np.diag(s)\n\nprint(f"A1 = U \u03a3 V\u1d40")\nprint(f"\\nU =\\n{np.round(U, 3)} (rotation\/reflection)")\nprint(f"\\n\u03a3 = diag{np.round(s, 3)} (scaling: \u03c31={s[0]:.3f}, \u03c32={s[1]:.3f})")\nprint(f"\\nV\u1d40 =\\n{np.round(Vt, 3)} (rotation\/reflection)")\nprint(f"\\nReconstructed: U @ \u03a3 @ V\u1d40 =\\n{np.round(U @ Sigma @ Vt, 6)}")\nprint(f"Original A1:\\n{A1} \u2713")<\/code><\/pre>\n\n\n\nA1 = U \u03a3 V\u1d40\n\nU =\n[[-0.851 -0.526]\n [-0.526 0.851]] (rotation\/reflection)\n\n\u03a3 = diag[2.309 1.191] (scaling: \u03c31=2.309, \u03c32=1.191)\n\nV\u1d40 =\n[[-0.851 -0.526]\n [-0.526 0.851]] (rotation\/reflection)\n\nReconstructed: U @ \u03a3 @ V\u1d40 =\n[[2. 0.5]\n [0.5 1.5]]\nOriginal A1:\n[[2. 0.5]\n [0.5 1.5]] \u2713<\/pre>\n\n\n\n# Visualise the four stages: v \u2192 V\u1d40v \u2192 \u03a3V\u1d40v \u2192 U\u03a3V\u1d40v\nfig, axes = plt.subplots(1, 4, figsize=(18, 5))\n\n# We'll track the unit circle and the standard basis arrows through each stage\ntheta = np.linspace(0, 2 * np.pi, 300)\ncircle = np.row_stack([np.cos(theta), np.sin(theta)])\n\nstages = [\n (np.eye(2), 'Stage 0: Input space\\n(unit circle + basis)'),\n (Vt, 'Stage 1: Apply $V^\\\\top$\\n(rotate to align with singular vectors)'),\n (Sigma @ Vt, 'Stage 2: Apply $\\\\Sigma$\\n(scale: stretch\/compress)'),\n (U @ Sigma @ Vt, 'Stage 3: Apply $U$\\n(rotate to output orientation = $A_1$)'),\n]\n\nfor ax, (M, title) in zip(axes, stages):\n draw_grid(ax, A=M if not np.allclose(M, np.eye(2)) else None)\n c = M @ circle\n ax.plot(c[0], c[1], color='steelblue', lw=2)\n ax.fill(c[0], c[1], color='steelblue', alpha=0.15)\n draw_arrow(ax, M[:, 0], color='crimson', lw=2)\n draw_arrow(ax, M[:, 1], color='darkorange', lw=2)\n style_ax(ax, title=title, xlim=(-3.5, 3.5), ylim=(-3.5, 3.5))\n\nfig.suptitle('SVD decomposition: $A_1 = U\\\\Sigma V^\\\\top$ \u2014 three geometric stages', fontsize=13)\nplt.tight_layout()\nplt.show()<\/code><\/pre>\n\n\n\n![\"\"](\"https:\/\/tensorzen.online\/wp-content\/uploads\/2026\/06\/image-13-1024x293.png\")
<\/figure>\n\n\n\n6. What Non-Singularity Guarantees<\/h2>\n\n\n\n
Bringing it all together:<\/p>\n\n\n\n
print("=== Non-singular matrix A1 ===")\nprint(f" Singular values : {np.round(s, 4)} (all > 0 \u2713)")\nprint(f" Determinant : {np.linalg.det(A1):.4f} (\u2260 0 \u2713)")\nprint(f" Rank : {np.linalg.matrix_rank(A1)} (= n \u2713)")\nprint(f" Condition number \u03ba : {s[0]\/s[-1]:.2f} (finite \u2713)")\nprint(f" Invertible : Yes")\n\nprint()\nprint("=== Singular matrix A2 ===")\nU2, s2, Vt2 = np.linalg.svd(A2)\nprint(f" Singular values : {np.round(s2, 6)} (one = 0 \u2717)")\nprint(f" Determinant : {np.linalg.det(A2):.6f} (\u2248 0 \u2717)")\nprint(f" Rank : {np.linalg.matrix_rank(A2)} (< n \u2717)")\nprint(f" Condition number \u03ba : \u221e")\nprint(f" Invertible : No \u2014 collapses 2D space onto a 1D line")<\/code><\/pre>\n\n\n\n=== Non-singular matrix A1 ===\n Singular values : [2.309 1.191] (all > 0 \u2713)\n Determinant : 2.7500 (\u2260 0 \u2713)\n Rank : 2 (= n \u2713)\n Condition number \u03ba : 1.94 (finite \u2713)\n Invertible : Yes\n\n=== Singular matrix A2 ===\n Singular values : [2.5 0. ] (one = 0 \u2717)\n Determinant : 0.000000 (\u2248 0 \u2717)\n Rank : 1 (< n \u2717)\n Condition number \u03ba : \u221e\n Invertible : No \u2014 collapses 2D space onto a 1D line<\/pre>\n\n\n\nSummary<\/h2>\n\n\n\nConcept<\/th> Geometric meaning<\/th><\/tr><\/thead> Active interpretation<\/td> $A$ moves vectors; coordinate system is fixed<\/td><\/tr> Passive interpretation<\/td> $A$ reshapes the grid; vectors are fixed<\/td><\/tr> Columns of $A$<\/td> Where the standard basis vectors $e_j$ land<\/td><\/tr> Non-singular<\/td> Columns independent \u2192 no dimension collapsed \u2192 full rank \u2192 invertible<\/td><\/tr> $\\det(A) \\neq 0$<\/td> Area\/volume is scaled but not destroyed<\/td><\/tr> $\\det(A) < 0$<\/td> Orientation is reversed (reflection component present)<\/td><\/tr> $\\det(A) = 0$<\/td> At least one dimension is collapsed \u2014 transformation irreversible<\/td><\/tr> SVD $A = U\\Sigma V^\\top$<\/td> Rotate \u2192 scale along axes \u2192 rotate<\/td><\/tr> All $\\sigma_i > 0$<\/td> Equivalent condition for non-singularity via SVD<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\nThe active and passive views are two lenses on the same object. Choosing between them is a matter of which makes the problem at hand easier to reason about \u2014 not a question of which is correct.<\/p>\n","protected":false},"excerpt":{"rendered":"
This notebook builds geometric intuition for matrix transformations, covering: 1. Active vs. Passive Interpretation Given a matrix $A$ and a vector $v$, the product $Av$ has two valid […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25,6,1],"tags":[],"class_list":["post-1007","post","type-post","status-publish","format-standard","hentry","category-in-english","category-matchematics","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/posts\/1007","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tensorzen.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1007"}],"version-history":[{"count":2,"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/posts\/1007\/revisions"}],"predecessor-version":[{"id":1021,"href":"https:\/\/tensorzen.online\/index.php?rest_route=\/wp\/v2\/posts\/1007\/revisions\/1021"}],"wp:attachment":[{"href":"https:\/\/tensorzen.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tensorzen.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tensorzen.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}